# Degeneration of $7$-dimensional minimal hypersurfaces which are stable or
have bounded index

Nick Edelen  Department of Mathematics, University of Notre Dame, Notre Dame,
IN, 46556<EMAIL_ADDRESS>
###### Abstract.

A $7$-dimensional area-minimizing embedded hypersurface $M^{7}$ will in
general have a discrete singular set, and the same is true if $M$ is locally
stable provided $\mathcal{H}^{6}(\mathrm{sing}M)=0$. We show that if
$M_{i}^{7}$ is a sequence of 7D minimal hypersurfaces which are minimizing,
stable, or have bounded index, then $M_{i}\to M$ can limit to a singular
$M^{7}$ with only very controlled geometry, topology, and singular set. We
show one can always “parameterize” a subsequence $i^{\prime}$ with controlled
bi-Lipschitz maps $\phi_{i^{\prime}}$ taking
$\phi_{i^{\prime}}(M_{1^{\prime}})=M_{i^{\prime}}$. As a consequence, we prove
the space of smooth, closed, embedded minimal hypersurfaces $M$ in a closed
Riemannian $8$-manifold $(N^{8},g)$ with a priori bounds
$\mathcal{H}^{7}(M)\leq\Lambda$ and $\mathrm{index}(M)\leq I$ divides into
finitely-many diffeomorphism types, and this finiteness continues to hold if
one allows the metric $g$ to vary, or $M$ to be singular.

##  1\. Introduction

We are interested in the space of $7$-dimensional embedded minimal
hypersurfaces which are area-minimizing, stable, or have bounded index.

An $n$-dimensional area-minimizing hypersurface $M^{n}$ will in general have a
$(n-7)$-dimensional singular set $\overline{M}\setminus M$, where by
$\overline{M}$ we denote set-theoretic closure. The same regularity holds if
$M$ is locally stable for the area-functional [SS81], [Wic14], provided one
knows a priori that $\mathcal{H}^{n-1}(\overline{M}\setminus M)=0$ (that is,
$M$ has no junctions modeled on half-planes meeting along an edge). A
significant goal in minimal surface theory is to understand the nature of the
singular set, and the structure of $M$ nearby.

Deep work of [Sim95], [NV20] have shown that the singular set of $M^{n}$ as
above is in fact countably-$(n-7)$-rectifiable, with locally-finite
$(n-7)$-dimensional area. Rectifiability is the best one can hope for, as
[Sim21b] has constructed remarkable examples of stable, embedded minimal
$(n\geq 8)$-dimensional hypersurfaces in smooth manifolds
$(\mathbb{R}^{n+1},g)$ with singular set being any arbitrary preassigned
closed subset of $\\{0^{8}\\}\times\mathbb{R}^{n-7}$. Moreover, at scale $1$
these minimal surfaces can be made to look arbitrarily varifold-close to
$(\text{Simons' cone})\times\mathbb{R}^{n-7}$, and the metric $g$ made to look
arbitrarily smoothly close to Euclidean.

The examples of [Sim21b] suggest there is little hope, for smooth metrics at
least, of understanding the fine-scale structure of an area-minimizing
hypersurface $M^{n}$ from only the knowledge of its tangent cones or its
behavior at large scales. Said differently, if $M_{i}$ is a sequence of area-
minimizing $(n\geq 8)$-dimensional hypersurfaces varifold-converging to some
$M$, then one cannot hope to understand the regular or singular structure of
$M_{i}$ based on $M$.

On the other hand, we shall show in this paper that these questions have
satisfactory answers in the “edge case” when $n=7$. (If $n\leq 6$ then area-
minimizing $M_{i},M$ are all regular, and convergence $M_{i}\to M$ is always
smooth.) In dimension $n=7$, the singular set is discrete, and any tangent
cone has a smooth link. Profound works of [AA81], [Sim83a] imply that near any
such isolated singular point, $M$ is a $C^{1}$ perturbation of its tangent
cone, giving a very precise fine-scale picture of $M$ at every point.

The Hardt-Simon foliation [HS85], [BDGG69] gives examples of smooth, entire,
area-minimizing hypersurfaces in $\mathbb{R}^{8}$ which are asymptotic to
singular cones. So, somewhat analogous to the $n\geq 8$ case, two area-
minimizing $7$-hypersurfaces that look very close at scale $1$ can have very
different regular or singular structure at scale $\epsilon$. However, in sharp
contrast to the higher-dimensional case, when $M^{7}$ resembles a Simons cone
$\mathbf{C}^{3,3}$ or $\mathbf{C}^{2,4}$ one can still give a precise
description of $M$: [ES19] (see also [SS86]) has shown that, besides the cone
itself, leaves of the foliation are the only other possible pictures for $M$.
In other words, if $M_{i}^{7}$ limit to a singular $M^{7}$, then near any
singularity of $M$ modeled on a minimizing quadratic cone, the $M_{i}$ are
locally $C^{1,\alpha}$ perturbations of either the cone, or a leaf of the
foliation.

Our goal is to prove a similar result for arbitrary stable $7$-dimensional
singularities. Precisely, we answer the following question:

######  Question 1.1.

Let $g_{i}\to g$ be a sequence of metrics on $B_{1}(0)\subset\mathbb{R}^{8}$,
and $M_{i}^{7}$ be a sequence of stable embedded minimal hypersurfaces in
$(B_{1},g_{i})$ with discrete singular set $\overline{M_{i}}\setminus M_{i}$,
which converge as varifolds in $B_{1}$ to some minimal surface $M$ in
$(B_{1},g)$. How can the geometry and topology of the $M_{i}$ degenerate near
a singular point of $M$?

Near any regular point of $M$, the sheeting theorem of [SS81] implies the
$M_{i}$ converge smoothly with possible multiplicity, and the a priori bound
of [NV20] implies that only finitely-many singular points of $M_{i}$ can
collapse into a singularity of $M$. So the interesting behavior is in the
regular structure of the $M_{i}$ as it collapses into a singular point of $M$
(we mention that our results do not rely on the estimates of [NV20], but
rather obtain [NV20]’s bound in dimension $7$ as a corollary of our main
theorem).

Aside from being interesting in its own right, understanding how a sequence of
minimal surfaces $M_{i}$ can behave as they approach a possibly singular limit
is important for several reasons. If one wants to answer quantitative or
global questions about moduli spaces of minimal surfaces (e.g. how many
diffeomorphism types are there, which geometric quantitaties are bounded, what
sort of compactness theory holds), one needs to know how the geometry of the
$M_{i}$ can degenerate along a limit. If one wants to answer questions about
the singular set (e.g. bounding the dimension or measure of certain strata),
one needs to know how these singularities can behave when passing to limits.

Even if one is attempting to understand the behavior of a _fixed_ minimal
surface $M^{n}$ near some singular point, if a tangent cone $\mathbf{C}$ at
that point itself has more than one singularity, then one is confronted with
variants of Question 1.1. For example, if $n=8$ and for some sequence of
dilations $r_{i}\to 0$, the rescaled $M_{i}:=(M-p)/r_{i}$ converge to
$\mathbf{C}=\mathbf{C}^{3,3}\times\mathbb{R}$, then near any point in
$\mathbf{C}$ away from the origin, the $M_{i}$ will resemble a largely
arbitrary sequence of minimal surfaces converging to $\mathbf{C}$.

We answer Question 1.1 in Theorem 2.1, where we show that the geometry,
topology, and singular set of the $M_{i}$ is precisely controlled near any
singular limit point of $M$. In fact we answer Question 1.1 assuming the
$M_{i}$ are only locally stable, but with a uniform bound on the index of the
stability operator. It turns out the techniques required to deal with stable
singularities extend with relatively little effort to deal with points of
index concentration.

We prove Theorem 2.1 by demonstrating a “cone-decomposition” for the $M_{i}$,
analogous to the bubble-tree decompositions for harmonic maps [SU81], [Par96],
or “neck-decompositions” for GH-limits of spaces with non-negative Ricci
[CN15], [JN21]: we show there is a finite, predetermined pool of “smooth
models” (entire minimal surfaces in $\mathbb{R}^{8}$ which are minimizing,
stable, or having bounded index), so that every $M_{i}$ breaks up into a
controlled number of annular “cone regions” wherein $M_{i}$ is a multi-graph
over some fixed stable cone, and a controlled number of “smooth regions”
wherein $M_{i}$ is a (scaled-down) multi-graph over one of the smooth models.

Unlike the finiteness theorems of [CKM17], [CN15], we have no classification
of tangent cone beyond being stable and with smooth cross section, and in
particular we cannot assume our cones are “integrable through rotations.” A
key technical issue we must overcome therefore is demonstrating that the cone
regions do not “rotate” through families of possibly pathological cones. We do
this in Sections 6, 11, using the Lojasiewicz-Simon inequality of [Sim83a].

Additionally, we show that, up to passing to a subsequence, we can find bi-
Lipschitz maps $\phi_{i}$ on the ambient space (e.g. the unit ball $B_{1}$),
which “parameterize” the $M_{i}$ in the sense that $\phi_{i}(M_{1})=M_{i}$,
and which have uniformly bounded Lipschitz constant. Due to the presense of
multiplicity, the construction of these maps (Section 8) turns out to be
fairly technical.

A consequence of our theorem is the following finitness result for minimal
hypersurface in an $8$-dimensional closed manifold. See Theorem 2.4 for a more
precise version, which additionally allows the metric to change and gives
quantitative bounds on the diffeomorphisms/bi-Lipschitz maps.

######  Theorem 1.2.

Let $(N^{8},g)$ be a closed, $8$-dimensional Riemannian manifold with $C^{3}$
metric $g$, and take $\Lambda,I\geq 0$. There is a number $K(N,g,I,\Lambda)$
so that every $C^{2}$, closed, embedded minimal hypersurface $M\subset(N,g)$
having $\mathcal{H}^{7}(M)\leq\Lambda$ and $\mathrm{index}(M)\leq I$ fits into
one of at most $K$ diffeomorphism classes. If $M$ is not closed but
$\overline{M}\setminus M$ is discrete, then $\overline{M}$ fits into one of at
most $K$ bi-Lipschitz equivalence classes.

The only known area-minimizing hypercones in $\mathbb{R}^{8}$ are the Simons’
cones $\mathbf{C}^{2,4}$ and $\mathbf{C}^{3,3}$, so for area-minimizing
$M_{i}^{7}$ it is conceivable that Question 1.1 can be entirely addressed
using the methods of [ES19]. On the other hand, the Simons’ cone
$\mathbf{C}^{1,5}\subset\mathbb{R}^{8}$ is known to be stable but not area-
minimizing (it is minimizing on only one side), and more generally [ES19]
cannot handle higher-multiplicity singularity models. To compare and contrast:
in [ES19] we address only multiplicity-one area-minimizing quadratic cones,
but we do so in any dimension, and we can give a priori $C^{1,\alpha}$ bounds.
In this paper, we can deal with any stable hypercone with possible
multiplicity, but only for $n=7$, and we can only give a priori $C^{1}$ bounds
on our maps.

In general there is not much progress on variants of Question 1.1 for singular
surfaces, in large part because (as illustrated by [Sim21b]’s examples) the
singular or regular nature of $M_{i}$ can drastically change in the limit. A
notable exception is when singularities of $M$ are modeled on a union of three
half-planes. This kind of singularity does not typically arise from the
“usual” minimization via integral currents or sets of locally-finite-
perimeter, but does arise naturally in so-called
$(\mathbf{M},\epsilon,\delta)$-minimizers. [Sim93] has shown that if $M_{i}\to
M$, and every singularity of $M$ has this type, then the $M_{i}$ are
$C^{1,\alpha}$ perturbations of $M$. These “$\mathbf{Y}$-type” singularities
are very special because they enforce the same singular structure on $M_{i}$.
In a similar vein, [CES22] showed the same regularity holds if $M$ has
“tetrahedral-type” singularities, and the $M_{i}$ secretly have an orientation
structure attached to them.

All the above mentioned results (like those in this paper) are local, allowing
for $M_{i}$ to be minimal in only a unit ball $(B_{1},g_{i})$, for some
sequence of metric $g_{i}$ close to Euclidean. Let us mention that [Wan20] has
proved a _global_ isolation theorem for certain $M$ in a (fixed) closed
$(N^{n+1},g)$. Analogous to [Whi94]’s results for strictly stable smooth
minimal surfaces, [Wan20] showed that if $M$ is strictly stable and has only
isolated singularities modelled on strictly-minimizing smooth cones, then $M$
is area-minimizing in some neighborhood $U$, and in fact is the unique
multiplicity-one minimal surface in $(U,g)$, so if $M_{i}\to M$ are all
minimal in $(N,g)$ then one must have $M_{i}=M$ for $i>>1$. (If $M$ is not
strictly stable, or some singularity is not strictly minimizing, then this
isolation can fail).

The past several years has seen significant study of the behavior of $(n\leq
6)$-dimensional embedded minimal hypersurfaces with bounded index (see e.g.
[Sha17], [CKM17], [BS18], [Car17], [Li17], [Son20]), motivated largely by
various min-max constructions. In this case, if $M_{i}\to M$ and the $M_{i}$
have uniformly bounded index $I$, then $M$ is entirely smooth, but convergence
may fail to be smooth at a collection of at most $I$ points of “index
concentration.” Question 1.1 in this setting has been very satisfactorily
answered by [CKM17], [BS18], using variants of a bubble-tree type
decomposition. [CKM17]’s approach is well-adapted for $M_{i}$ without area
bounds, while [BS18] gives additional information on total curvature and
curvature concentration. Most of the methods in these works rely very heavily
on the planar nature of singularities and smoothness of the limit. Our
theorems specialize to the $n\leq 6$ case, and give an alternate method of
proof to several of the results in [CKM17], [BS18], but we emphasize that the
real interest in this work is for singular $M$.

Singular $7$-dimensional minimal hypersurfaces with bounded index arise
naturally in both Almgren-Pitts ([Li20]) and Allen-Cahn ([Hie18], [Gas20])
min-max constructions. By carefully understanding the behavior of the min-max
surface and sweepout near its singular points, [CLS20] and [LW20] have
recently shown that for a generic metric on a closed $N^{8}$ there is a
smooth, closed, embedded minimal hypersurface. In a similar vein, there are
numerous perturbative existence results, which state that given a suitable
minimizing or stable $M^{7}\subset(N^{8},g)$, then by slightly perturbing the
boundary of $M$ ([HS85]) or the ambient metric $g$ ([Sma93], [Wan20]), one can
obtain a smooth $7$-dimensional minimizing or stable hypersurface.

[Whi91] has shown that for a closed manifold $N$, the space of ordered pairs
$(g,M)$ consisting of $C^{k}$ metrics $g$ on $N$, and $C^{j,\alpha}$ minimal
surfaces $M$ in $(N,g)$, can be given a Banach manifold structure, and
moreover that the projection $(g,M)\to g$ is Fredholm with index $0$.
[Whi91]’s theorem implies some remarkable genericity and finiteness results,
for example: for a generic “bumpy” $C^{3}$ metric $g$ on $N^{7}$ (in the sense
of Baire) there exist only finitely-many stable hypersurfaces
$M^{6}\subset(N,g)$ with $\mathcal{H}^{6}(M)\leq\Lambda$. White’s “bumpy”
metric theorem is a major ingredient in many generic min-max results.

A natural question garnering recent interest is whether a similar manifold
structure holds on the spaces of pairs of metrics and _possibly singular_
minimal surfaces. [Wan20] has made some headway on this question, by
developing a robust linear theory for Jacobi fields on minimal hypersurfaces
with isolated singularities, and using this to show that for certain
$M^{n}\subset(N^{n+1},g)$ (having bounded index, only strongly-isolated,
strictly-minimizing singularities, and a few other technical conditions),
every $C^{4}$ metric peturbation $g^{\prime}$ admits a minimal surface
$M^{\prime}\subset(N,g^{\prime})$ which is close to $M$.

One can view our Theorem 2.4 as a step towards the manifold-structure problem
from the opposite perspective, by showing that nearby metric-surface pairs
$(g^{\prime},M^{\prime 7})$ with controlled index in a closed $N^{8}$ can be
“parameterized” using finitely-many model surfaces. It would be very
interesting if bumpiness results similar to those of [Whi91] held in the
singular setting also, e.g. if for a generic $C^{3}$ metric $g$ on a closed
$N^{8}$, there are only finitely-many area-minimizing or stable hypersurfaces
$M^{7}\subset(N,g)$ satisfying $\mathcal{H}^{7}(M)\leq\Lambda$ and
$\dim(\mathrm{sing}M)=0$.

Finally, let us mention that the basic ideas that go into proving the main
Theorem 7.1 are fairly general, relying only on good compactness and partial
regularity theorems, a monotonicity formula, and the existence of a
Lojasiewicz-Simon-type inequality for singular models. One could likely adapt
the method of this paper to prove (for example) that there are only finitely-
many isotopy classes of smooth energy-minimizing maps
$(M^{3},g_{M})\to(N^{n},g_{N})$ having energy $\leq\Lambda$, where $M^{3}$ is
a closed $3$-dimensional manifold with smooth metric $g_{M}$, and $N^{n}$ a
closed $n$-dimensional manifold with analytic metric $g_{N}$.

Acknowledgements: I am indebted to Otis Chodosh, Luca Spolaor, Gabor
Szekelyhidi, and Brian White for many useful discussions, and to Andrew Putman
for patiently answering my naive questions about mapping class groups. Part of
this work was conceived while visiting Stanford University and Big Basin State
Park. I thank them both for their hospitality, and wish Big Basin a speedy
recovery.

##  2\. Main theorems

To more easily state our results we require a little notation. Let $(N^{8},g)$
a Riemannian manifold with $C^{3}$ metric $g$. Since many of our results
involve changing the metric, we will typically explicitly specify the metric
dependence. Let us write $\mathcal{H}^{7}_{g}$ for the $7$-dimensional
Hausdorff measure in $(N,g)$, $\mathrm{Ric}_{g}$ for the Ricci curvature,
$B_{r}^{g}(a)$ for the open geodesic ball centered at $a$, and $d_{H,g}$ for
the Hausdorff distance. If $\phi:N\to N$, we write $\mathrm{Lip}_{g}(\phi)$
for the Lipschitz constant of $\phi$ w.r.t. $g$. If $U\subset N$, then
$\overline{U}$ denotes set-theoretic closure.

If $N=U\subset\mathbb{R}^{8}$, all measurements, derivatives, norms, etc. will
be taken w.r.t. the Euclidean metric $g_{\mathrm{eucl}}$ unless explicitly
stated otherwise. So, if $N\subset\mathbb{R}^{8}$, $\mathcal{H}^{7}$ will
always denote the Euclidean Hausdorff measure, $B_{r}(a)$ the Euclidean open
$r$-ball centered at $a$, $A_{R,r}(a)=B_{R}(a)\setminus\overline{B_{r}(a)}$
for the open annulus, $d_{H}$ the Euclidean Hausdorff distance, and
$\mathrm{Lip}(\phi)$ for the Lipschitz constant w.r.t. $g_{\mathrm{eucl}}$.
Similarly, if $M$ is a $C^{1}$ embedded submanifold of $\mathbb{R}^{8}$, we
write $T^{\perp}M$ for the Euclidean normal bundle of $M$, $u:M\to M^{\perp}$
for a section of this normal bundle, and $\nabla$ the connection induced by
the Euclidean connection on $T^{\perp}M$.

If $M$ is a $C^{1}$ embedded minimal hypersurface of $(N,g)$, define
$\mathrm{reg}M$ to be the set of points $x\in\overline{M}$ having the property
that in some open neighborhood $U$ of $x$, $\overline{M}\cap U$ is a closed
(as sets) embedded submanifold of $U$. Define
$\mathrm{sing}M=\overline{M}\setminus\mathrm{reg}M$. Of course
$\mathrm{sing}M\subset\overline{M}\setminus M$, but there could be points in
$\overline{M}\setminus M$ that “secretly” are regular points, and we want to
make the notion of singular set canonical. There is no loss in always assuming
that $M=\mathrm{reg}M$.

Let us write $T^{\perp_{g}}M$ for the normal bundle of $M$ with respect to
$g$, and $u:M\to M^{\perp_{g}}$ for a section of this normal bundle, and
$\nabla^{g}$ the connection induced by $g$ on the normal bundle. For $M$ being
$C^{2}$, write $A_{M,g}$ for the second fundamental form of $M\subset(N,g)$.
We define $\mathrm{index}(M,N,g)$ to be the maximal dimension of subspace
$\subset C^{1}_{c}(M,M^{\perp_{g}})$ on which the quadratic form

(1)
$Q_{M,g}(u,u)=\int_{M}|\nabla^{g}u|^{2}-|A_{M,g}|^{2}|u|^{2}-\mathrm{Ric}_{g}(u,u)d\mathcal{H}^{7}_{g}$

is strictly negative definite. If $M$ is stable for the area functional in
$(N,g)$, then $Q_{M,g}\geq 0$.

Define $\mathcal{M}_{7}(N^{8},g)$ to be the space of $C^{2}$ embedded minimal
hypersurfaces in $(N,g)$ with the property that $\overline{M}\setminus M$
consists of a discrete set of points. It follows by work of [SS81], [Wic14]
that if $M$ is any $C^{2}$ embedded minimal hypersurface in $(N,g)$ satisfying
$\mathrm{index}(M,N,g)<\infty$ and $\mathcal{H}^{6}(\overline{M}\setminus
M)=0$, then $M\in\mathcal{M}_{7}(N,g)$. See Section 4 for more details. We
define $\mathcal{C}$ to be the set of smooth (away from $0$), stable minimal
hypercones in $\mathbb{R}^{8}$.

If $M$ is a $C^{1}$ $n$-dimensional embedded hypersurface in
$\mathbb{R}^{n+1}$, we define
$\theta_{M}(a,r)=\omega_{n}^{-1}r^{-n}\mathcal{H}^{n}(M\cap B_{r}(a))$ for the
density ratio of $M$. More generally, if $V$ is an $n$-varifold with mass
measure $\mu_{V}$, then
$\theta_{V}(a,r)=\omega_{n}^{-1}r^{-n}\mu_{V}(B_{r}(a))$. If $M=\mathbf{C}$ is
a cone, then we write $\theta_{\mathbf{C}}(0)=\theta_{\mathbf{C}}(0,1)$. If
$M$ is a $C^{1}$ embedded hypersurface in $(N^{8},g)$, write $[M]_{g}$ for the
$7$-varifold obtained by integrating over $M$, i.e. with mass measure
$\mu_{[M]_{g}}=\mathcal{H}^{7}_{g}\llcorner M$.

Our main result is local, and has two parts: first, Theorem 7.1, which builds
a “cone decomposition” for a minimal surface $M$ with bounded index and area
sufficiently nearby singular cone; and second, Theorem 8.1, which
“parameterizes” minimal surfaces admitting cone decompositions. Together, they
say:

######  Theorem 2.1.

Given $\Lambda,I\geq 0$, and $\sigma,\beta\in(0,\frac{1}{10^{3}(I+1)})$, then
there is a finite collection of “$(\Lambda,\beta,\sigma)$-smooth models”
$\mathcal{S}$ (Definition 7.0.1), and constants $K$, $\delta>0$, all depending
only on $(\Lambda,I,\sigma,\beta)$, so that the following holds.

Take $\mathbf{C}\in\mathcal{C}$ so that $\theta_{\mathbf{C}}(0)\leq\Lambda$.
Let $g_{i},g$ be $C^{3}$ metrics on $B_{1}(0)$ such that
$|g_{i}-g_{\mathrm{eucl}}|_{C^{3}(B_{1})}\leq\delta$, $g_{i}\to g$ in
$C^{2}(B_{1})$, and let $M_{i}\in\mathcal{M}_{7}(B_{1},g_{i})$ be a sequence
of minimal surfaces satisfying

$d_{H}(M_{i}\cap B_{1},\mathbf{C}\cap
B_{1})\leq\delta,\quad\mathcal{H}^{7}_{g_{i}}(M_{i}\cap
B_{1})\leq\Lambda,\quad\mathrm{index}(M_{i},B_{1},g_{i})\leq I.$

Then, after passing to a subsequence, we can find a radius
$r\in(1-200\sigma(I+1),1)$, so that every $M_{i}\cap(B_{r}(0),g_{i})$ admits a
“$(\Lambda,\beta,\mathcal{S},K)$-cone decomposition” (Definition 7.0.4).

Moreover, passing to a further subsequence, we can find bi-Lipschitz maps
$\phi_{i}:B_{r}\to B_{r}$ and a constant $C$ so that:

  1. (1)

$\phi_{i}(\overline{M_{1}}\cap B_{r})=\overline{M_{i}}\cap B_{r}$, and
$\phi_{i}(\mathrm{sing}M_{1}\cap B_{r})=\mathrm{sing}M_{i}\cap
B_{r}=\mathrm{sing}M_{i}\cap B_{(1-\sigma)r}$;

  2. (2)

$\phi_{i}$ restricts to a $C^{2}$ diffeomorphism
$B_{r}\setminus\mathrm{sing}M_{1}\to B_{r}\setminus\mathrm{sing}M_{i}$;

  3. (3)

$\mathrm{Lip}(\phi_{i})\leq C$;

  4. (4)

there is a stationary integral varifold $V$ in $(B_{1},g)$ of the form
$V=m_{1}[M_{1}^{\prime}]_{g}+\ldots+m_{k}[M_{k}^{\prime}]_{g}$, for
$M_{i}^{\prime}\in\mathcal{M}_{7}(B_{1},g)$ and $m_{i}\in\mathbb{N}$, so that
$[M_{i}]_{g_{i}}\to V$ as varifolds in $B_{1}$, and in $C^{2}$ on compact
subsets of $B_{1}\setminus(\mathrm{sing}V\cup\mathcal{I})$ for some set
$\mathcal{I}$ of at most $I$ points;

  5. (5)

there is a Lipschitz $\phi_{\infty}:B_{r}\to B_{r}$ such that
$\phi_{\infty}(\overline{M_{1}}\cap B_{r})=\mathrm{spt}V\cap B_{r}$ and
$\phi_{i}\to\phi_{\infty}$ in $C^{\alpha}(B_{r})$ for all $\alpha\in(0,1)$.

######  Remark 2.2.

In certain circumstances (when singularities of the $M_{i}$ are “integrable
through rotations”) then one can arrange each $\phi_{i}$ to be globally
$C^{1,\alpha}$ for some $\alpha>0$. Ditto for Theorems 2.4, 2.6. See Remarks
8.2, 8.8.

######  Remark 2.3.

Theorem 2.1 (and Theorems 2.4, 2.6) also hold for smooth, $n$-dimensional
minimal hypersurfaces, when $2\leq n\leq 6$. In this case we additionally
obtain the curvature bounds

(2) $\int_{M_{i}\cap B_{r}}|A_{M_{i},g_{i}}|^{n}d\mathcal{H}^{n}_{g_{i}}\leq
C$

for $C$ depending only on $(\Lambda,I,\beta,\sigma)$ (recovering the total
curvature bound of [BS18]). (2) is a consequence of a more general Dini-type
estimate, see Remark 6.7

There are two key ideas that go into proving Theorem 2.1 (or, rather, Theorem
7.1), both heavily indebted to the Lojasiewicz-Simon inequality [Sim83a]. The
first is that the set of densities for smooth, embedded, stable minimal cones
in $\mathbb{R}^{8}$ forms a discrete set. This allows us to prove Theorem 7.1
by induction on $\Lambda$, reducing to one of the base cases of either
$\mathbf{C}$ being a multiplicity-one plane (in which case the result follows
by Allard’s theorem [All72]), or $\mathbf{C}$ being a plane-with-multiplicity
and the $M_{i}$ being stable (in which case the result follows by Schoen-Simon
[SS81]).

The second key idea is captured in Theorem 6.3, where we show that if a
minimal surface $M$ in $B_{R}(a)$ has very small density drop in some annulus
$B_{R}(a)\setminus B_{\rho}(a)$, then in this annulus $M$ is graphical over
some fixed cone, _irrespective_ of what $M$ looks like inside $B_{\rho}(a)$.
We need this estimate to rule out the cones “rotating” in annular regions
obtained from Theorem 7.1. To prove Theorem 6.3 we adapt the decay/growth
estimates of [Sim83a], which loosely say that a non-conical minimal graph over
a cone must quantitatively decay or growth either polynomially or
logarithmically from one scale to the next.

The third, let’s say key half-idea, involved in Theorem 2.1 is in constructing
the parameterizations $\phi_{i}$ (Theorem 8.1). The construction turns out to
be fairly technical, because the presence of multiplicity means that the
$\phi_{i}$ will only be $C^{0}$ perturbations of the identity, rather than
$C^{1}$ perturbations. So one cannot use “naive” gluing methods to piece
together the maps, but must instead use very strongly the exact structure of
the $\phi_{i}$.

Theorem 2.1 implies the following global finiteness theorem in a closed
Riemannian $8$-manifold.

######  Theorem 2.4.

Let $(N^{8},g)$ be a closed $8$-dimensional Riemannian manifold with $C^{3}$
metric $g$. Given $\Lambda,I\geq 0$, there is a number
$\delta(N,g,\Lambda,I)>0$ so that the following holds.

Let $\mathcal{G}$ be any set of $C^{3}$ metrics on $N$ satisfying
$|g-g^{\prime}|_{C^{3}(N,g)}\leq\delta$ for every $g^{\prime}\in\mathcal{G}$.
Then we can find a constant $C(N,g,\Lambda,I,\mathcal{G})$, and _finite_
collections of metrics $\\{g_{v}\\}_{v}\subset\mathcal{G}$, and minimal
surfaces $\\{M_{v}\in\mathcal{M}_{7}(N,g_{v})\\}_{v}$, with the properties:

$\mathcal{H}^{7}_{g_{v}}(M_{v})\leq\Lambda,\quad\mathrm{index}(M_{v},N,g_{v})\leq
I,$

so that if $M\in\mathcal{M}_{7}(N,g^{\prime})$ for some
$g^{\prime}\in\mathcal{G}$ satisfies
$\mathcal{H}^{7}_{g^{\prime}}(M)\leq\Lambda$,
$\mathrm{index}(M,N,g^{\prime})\leq I$, then there is a $v$ and a bi-Lipschitz
mapping $\phi:(N,g)\to(N,g)$ such that:

  1. (1)

$\phi(\overline{M_{v}})=\overline{M}$,
$\phi(\mathrm{sing}M_{v})=\mathrm{sing}M$;

  2. (2)

$\phi$ restricts to a $C^{2}$ diffeomorphism $N\setminus\mathrm{sing}M_{v}\to
N\setminus\mathrm{sing}M$;

  3. (3)

$\mathrm{Lip}_{g}(\phi)\leq C$.

If $M$ is area-minimizing, then one can assume $M_{v}$ is area-minimizing too.

######  Remark 2.5.

[Wan20] has shown that if $(N^{8},g)$ is closed and $g$ is $C^{4}$, then given
any $\epsilon>0$ and any $M\in\mathcal{M}_{7}(N,g)$ having bounded index which
is orientable, “non-degenerate,” and with the property that every singular
point is modelled on a strictly stable and strictly minimizing cone (e.g. a
minimizing Simons’ cone), then there is a $C^{4}$ neighborhood of metrics
$\mathcal{G}\ni g$ so that for each $g^{\prime}\in\mathcal{G}$, one can find
an $M^{\prime}\in\mathcal{M}_{7}(N,g^{\prime})$ with bounded index satisfying
$d_{H,g}(M,M^{\prime})\leq\epsilon$.

We also have a corresponding finiteness result in Euclidean space.

######  Theorem 2.6.

Given $\Lambda,I\geq 0$, we can find a finite collection
$\\{M_{v}\\}_{v}\subset\mathcal{M}_{7}(\mathbb{R}^{8},g_{\mathrm{eucl}})$, and
an increasing function $C_{\Lambda,I}:\mathbb{R}\to\mathbb{R}$, with the
properties

(3)
$\theta_{M_{v}}(0,\infty)\leq\Lambda,\quad\mathrm{index}(M_{v},\mathbb{R}^{8},g_{\mathrm{eucl}})\leq
I,$

so that given any other
$M\in\mathcal{M}_{7}(\mathbb{R}^{8},g_{\mathrm{eucl}})$ satisfying (3), we can
find a number $\lambda>0$, a locally bi-Lipschitz map
$\phi:\mathbb{R}^{8}\to\mathbb{R}^{8}$, and a $v$, so that

(4)
$\displaystyle\phi(\overline{M_{v}})=\lambda\overline{M},\quad\phi(\mathrm{sing}M_{v})=\lambda\mathrm{sing}M_{v},$
(5) $\displaystyle\phi|_{\mathbb{R}^{8}\setminus\mathrm{sing}M_{v}}\text{ is a
$C^{2}$ diffeomorphism},$ (6) $\displaystyle\text{and
}\mathrm{Lip}(\phi|_{B_{r}(0)})\leq C_{\Lambda,I}(r)\text{ for every $r$}.$

If $M$ is area-minimzing (resp. is an area-minimizing boundary), we can assume
$M_{v}$ is area-minimizing (resp. is an area-minimizing boundary). If the
tangent cone of $M$ at infinity is multiplicity-one (e.g. as when $M$ is an
area-minimizing boundary), then $\mathrm{Lip}(\phi)\leq C=C_{\Lambda,I}(10)$
on all of $\mathbb{R}^{8}$.

######  Remark 2.7.

[Cha97] (see also [Whi89]) has shown that if $\mathbf{C}^{7}\in\mathcal{C}$ is
strictly-minimizing, then for every “slowly-decaying Jacobi field” $v$ of
$\mathbf{C}$ there is a area-minimizing hypersurface $M\subset\mathbb{R}^{8}$
which is asymptotic to $\mathrm{graph}_{\mathbf{C}}(v)$. In particular, if
$I=\mathrm{index}(\mathbf{C}\cap\partial B_{1},S^{7},g_{\mathrm{round}})$,
this gives an $I$-parameter family of distinct, area-minimizing hypersurfaces
asymptotic to $\mathbf{C}$. Theorem 2.6 implies that this family has bounded
geometry and topology.

Lastly, let us mention that our smallness estimate of Theorem 6.3 allows us to
sharpen the Corollary [ES19, Corollary 3.4]. For a minimal surface $M$ in a
non-Euclidean metric, in [ES19] we could not rule out the possibility that at
different radii $M$ resembled different rotations of the Simons’ cone (which
was still sufficient to characterize singular nature of $M$). Though we still
cannot get the effective $C^{1,\alpha}$ estimates of [ES19, Theorem 3.1] in
Euclidean space, here we can at least show the cones do not rotate. See [ES19]
for more details on background and notation. The proof of Theorem 2.8 (given
in Section 13) is something of a baby version of the proof of our main Theorem
7.1, and in fact it might be helpful for the reader to read this proof first.

######  Theorem 2.8.

Take any $n\in\mathbb{N}$, and let $\mathbf{C}^{n}\subset\mathbb{R}^{n+1}$ be
a minimizing quadratic hypercone. Let $\\{S_{\lambda}\\}_{\lambda}$ be the
leaves of the Hardt-Simon foliation of $\mathbf{C}$, so that
$S_{0}=\mathbf{C}$. Given any $\epsilon>0$, there is a
$\delta(\mathbf{C},\epsilon)$ so that the following holds. Let $g$ be a
$C^{3}$ metric on $B_{1}\subset\mathbb{R}^{n+1}$ satisfying
$|g-g_{\mathrm{eucl}}|_{C^{3}(B_{1})}\leq\delta$. Let $V$ be an integral
stationary $n$-varifold in $(B_{1},g)$ satisfying

(7) $\displaystyle d_{H}(\mathrm{spt}V\cap B_{1},\mathbf{C}\cap
B_{1})\leq\delta,\quad(1/2)\theta_{\mathbf{C}}(0)\leq\theta_{V}(0,1/2),\quad\theta_{V}(0,1)\leq(3/2)\theta_{\mathbf{C}}(0).$

Then we can find an $a\in\mathbb{R}^{n+1}$, $\lambda\in\mathbb{R}$, $q\in
SO(n+1)$, satisfying

(8) $|a|+|q-Id|+|\lambda|\leq\epsilon,$

and a $C^{2}$ function $u:(a+q(S_{\lambda}))\cap B_{1/2}(a)\to
S_{\lambda}^{\perp}$ such that

(9) $\mathrm{spt}V\cap B_{1/4}=\mathrm{graph}_{a+q(S_{\lambda})}(u)\cap
B_{1/4},\quad|x-a|^{-1}|u|+|\nabla u|+|x-a||\nabla^{2}u|\leq\epsilon.$

If $\mathbf{C}^{n}\subset\mathbb{R}^{n+1}$ is a general smooth (away from $0$)
strictly-stable and strictly-minimizing hypercone, then the same conclusion
holds if one additionally assumes $\mathrm{spt}V$ lies to one side of
$\mathbf{C}$.

##  3\. Outline of proof

The key to our argument is the observation (by no means our own) that the set
of densities of smooth stable minimal $7$-dimensional hypercones is a discrete
set, which a direct consequence of the Lojasiewicz-Simon inequality [Sim83a]
and standard compactness/regularity theory for stable minimal hypersurfaces.
At some level this is saying that the “complexity” of the singular models
$\mathbf{C}$ is quantized. We prove our main decomposition Theorem 2.1 by
induction on the density (including multiplicity) of the cone $\mathbf{C}$.

Our basic idea is very simple. Unfortunately (fortunately?) there is a highly
non-trivial complication, due to the nature of “cone regions,” but this point
is not immediately obvious so let us ignore it for the moment. Let us
illustrate our idea by proving that if $M_{i}$ is a sequence of smooth, area-
minimizing boundaries in $B_{1}$ converging to an area-minimizing cone
$\mathbf{C}$, then after passing to a subsequence all the $M_{i}\cap B_{1/2}$
are diffeomorphic to each other. (So, everything is mulitplicity-one and
converges smoothly on the regular part).

Let us suppose, towards a contradiction, that none of the $M_{i}\cap B_{1/2}$
are diffeomorphic. Write $1=\theta_{0}<\theta_{1}<\ldots$ for the densities of
area-minimizing cones, and suppose $\theta_{\mathbf{C}}(0)=\theta_{k}$. If
$k=0$ then $\mathbf{C}$ is the plane, and so our assertion follows by Allard’s
theorem. Let us assume by induction our assertion holds for every
$\theta_{\mathbf{C}}(0)\leq\theta_{k-1}$.

The $M_{i}$ converge smoothly to $\mathbf{C}\cap B_{1}$ away from $0$. For any
$a\in B_{1/2}$ we can consider the smallest radius $\rho$ so that
$A_{1/2,\rho}(a)\cap M_{i}$ is a “$\mathbf{C}$-cone region.” Let’s be
intentionally vague for the moment, and just think of this as an annulus where
$M_{i}$ looks very close to a cone, which may be $\mathbf{C}$ or may be some
other $\mathbf{C}^{\prime}$ diffeomorphic to $\mathbf{C}$.

Let us pick $\rho_{i}$ to be the least such radius among all possible $a\in
B_{1/2}$, and pick $a_{i}$ realizing this point. Smooth convergence to
$\mathbf{C}$ on compact subsets of $B_{1}\setminus\\{0\\}$ implies
$\rho_{i}\to 0$ and $a_{i}\to 0$. Since all the $M_{i}\cap
A_{1/2,\rho_{i}}(a_{i})$, being $\mathbf{C}$-cone regions, are diffeomorphic
to $\mathbf{C}\cap A_{1,1/2}$, our contradiction hypothesis implies that none
of the $M_{i}\cap B_{2\rho_{i}}(a_{i})$ are diffeomorphic.

Now consider the rescaled surfaces
$M_{i}^{\prime}=\rho_{i}^{-1}(M_{i}-a_{i})$. After passing to a subsequence,
we get convergence $M_{i}^{\prime}\to M^{\prime}$, an area-minimizer in
$\mathbb{R}^{8}$. By construction, $M^{\prime}\cap A_{\infty,1}(a)$ is a
$\mathbf{C}$-cone region, and so $M^{\prime}$ is smooth outside $B_{1}$. Any
tangent cone at infinity of $M^{\prime}$ is a cone $\mathbf{C}^{\prime}$ very
near to $\mathbf{C}$, having the same density
$\theta_{\mathbf{C}^{\prime}}(0)=\theta_{\mathbf{C}}(0)$.

On the other hand $M^{\prime}$ cannot be smooth, because then
$M_{i}^{\prime}\to M^{\prime}$ smoothly on compact sets, contradicting the
fact that none of the $M_{i}^{\prime}\cap B_{2}$ are diffeomorphic. So there
are (finitely-many) singularities of $M^{\prime}$ in $B_{1}$. I claim that any
such singularity $x$ has density
$\theta_{M^{\prime}}(x)<\theta_{\mathbf{C}}(0)$. Otherwise, if we had
$\theta_{M^{\prime}}(x)\geq\theta_{\mathbf{C}}(0)$, then monotonicity would
imply that $M^{\prime}=x+\mathbf{C}^{\prime}$. But then, for $i$ large each
$M_{i}\cap A_{1/2,\rho_{i}/2}(a_{i}+\rho_{i}x)$ would be a $\mathbf{C}$-cone
region, contradicting our choice of $\rho_{i}$ as the least possible such
radius.

So in fact every singularity of $M^{\prime}$ has density $\leq\theta_{k-1}$.
In a small ball $B_{r_{x}}(x)$ around every singularity $x$, $M^{\prime}$
looks scale-invariantly close to a cone $\mathbf{C}_{x}$ having density
$\leq\theta_{k-1}$, and so by induction (if we take $r$ sufficiently small and
pass to a subsequence in $i$), all the $M_{i}^{\prime}\cap B_{r_{x}}(x)$ are
diffeomorphic. Since $M_{i}^{\prime}$ converge smoothly to $M^{\prime}$ away
from the singular set, we deduce all the $M_{i}^{\prime}\cap B_{2}$ are
diffeomorphic. This is a contradiction.

OK so what’s wrong with that argument. The catch is in exactly how we can
define a “cone region,” and how these are glued to the other parts of $M_{i}$.
When we do this contradiction argument, without using any additional
quantitative estimates, we can only pass information between finitely-many
scales, e.g. between scale $\rho$ and $\rho/2$. You will find that, no matter
how you define cone region (by density drop, by graphicality, etc.), you will
have to allow the model cone to change with the radius. This argument will
only show that for a given $M=M_{i}$, for every radius $r\in[\rho_{i},1/2]$,
there is a cone $\mathbf{C}_{r}$ diffeomorphic to $\mathbf{C}$ (in the
sphere), and having the same density, so that $M\cap A_{r,r/8}(a_{i})$ is a
small graph over $a_{i}+\mathbf{C}_{r}$. In certain special cases, such as
when $\mathbf{C}$ is planar with possible multiplicity ([Whi18], [BS18]) or
strictly-minimizing with multiplicity-one ([ES19], [Wan20]), one can rule out
the $\mathbf{C}_{r}$ changing, since there are good barriers which pass
information across scales by “trapping” $M$ to stay close to the original
$\mathbf{C}$. For more general $\mathbf{C}$ this is not (to my knowledge)
available.

The changing cones starts to becomes a problem because, although the
$M_{i}\cap A_{1,\rho_{i}}(a_{i})$ are still all diffeomorphic to an annulus
$\mathbf{C}\cap A_{1,1/2}$, we lose control over how this annulus is glued
into the region $M_{i}\cap B_{\rho_{i}}(a_{i})$. Actually, for our one single
step of induction, this is not strictly speaking an issue, because we could
push all the “rotation” of the cones out to scale $\approx 1$. Once you start
attempting to induct however, this is a problem, since we need to be able to
glue the cone regions at both ends.

Precisely, we require the following. Let $\phi_{ij}:M_{i}\cap
A_{1/2,\rho_{i}}(a_{i})\to M_{j}\cap A_{1/2,\rho_{j}}(a_{j})$ be our
diffeomorphisms. By our convergence and our choice of subsequence, there is a
fixed cone $\mathbf{C}^{\prime}$ diffeomorphic to $\mathbf{C}$ so that all the
$M_{i}\cap A_{1,1/2}$ are small graphs over $\mathbf{C}\cap A_{1,1/2}$, and
all the $M_{i}^{\prime}\cap A_{2,1}$ are small graphs over
$\mathbf{C}^{\prime}\cap A_{2,1}$, and therefore we obtain diffeomorphisms
$\psi_{i}:\mathbf{C}\cap A_{1,1/2}\to M_{i}\cap A_{1,1/2}$,
$\psi_{i}^{\prime}:\mathbf{C}^{\prime}\cap A_{2,1}\to M_{i}^{\prime}\cap
A_{2,1}$, which are all very small perturbations of the identity. To
adequately construct diffeomorphisms from $M_{i}$ to $M_{j}$, we require that
all the

$\psi_{j}^{-1}\circ\phi_{ij}\circ\psi_{i}:\mathbf{C}\cap
A_{1,1/2}\to\mathbf{C}\cap A_{1,1/2}$

are isotopic, _and_ all the

$\psi_{j}^{\prime-1}\circ\eta_{a_{j},\rho_{j}}\circ\phi_{ij}\circ\eta_{a_{i},\rho_{i}}^{-1}\circ\psi_{i}^{\prime}:\mathbf{C}^{\prime}\cap
A_{2,1}\to\mathbf{C}^{\prime}\cap A_{2,1}$

are isotopic, where we write $\eta_{x,r}(z)=(z-x)/r$.

Let us illustrate this with two (would-be) examples. In the first, take
$\mathbf{C}$ a hypercone in $\mathbb{R}^{4}$, and identify
$S^{3}\setminus\\{pt\\}$ with $\mathbb{R}^{3}$. Think of the link
$\mathbf{C}\cap\partial B_{1}$ as a genus-two surface in $\mathbb{R}^{3}$,
that looks like the number $8$ speared onto the $z$-axis. Suppose there is a
smooth family of smooth, minimal cones $\mathbf{C}_{t}$ for $t\in\mathbb{R}$,
so that as $t$ increases the cross section $\mathbf{C}_{t}\cap\partial B_{1}$
keeps one handle fixed, and rotates counter-clockwise the other handle around
the $z$-axis, and does so periodically in the sense that
$\mathbf{C}_{t+i}=\mathbf{C}_{t}$ for every integer $i$.

Consider now the surface $M$ defined by $M\cap\partial
B_{r}=\mathbf{C}_{-\sigma\log r}\cap\partial B_{r}$ for some small $\sigma>0$,
so that $M\cap A_{1,0}$ is a $\mathbf{C}$-cone region. The annuli
$M_{i}:=M\cap A_{1,e^{-i/\sigma}}$ are all diffeomorphic, and in fact
$(e^{i/\sigma}M_{i})\cap A_{1,1/2}=(e^{j/\sigma}M_{j})\cap A_{1,1/2}$ for
integers $i$, $j$, but for any two diffeomorphisms $\phi_{ij}:M_{i}\to M_{j}$,
$\phi_{ij^{\prime}}:M_{i}\to M_{j^{\prime}}$ (with $j\neq j^{\prime}$) which
are isotopic as maps $A_{1,1/2}\cap M_{i}\to A_{1,1/2}\cap M_{j}\equiv
A_{1,1/2}\cap M_{j^{\prime}}\equiv A_{1,1/2}\cap M_{1}$, the rescaled
diffeomorphisms

$\eta_{0,e^{-j/\sigma}}\circ\phi_{ij}\circ\eta_{0,e^{-i/\sigma}}^{-1},\eta_{0,e^{-j^{\prime}/\sigma}}\circ\phi_{ij^{\prime}}\circ\eta_{0,-i/\sigma}^{-1}:M_{1}\cap
A_{1,1/2}\to M_{1}\cap A_{1,1/2}$

cannot also be isotopic.

The second example is simpler, but avoidable. If the $M_{i}$ had only bounded
index, then $\mathbf{C}$ could be a multiplicity-two plane. As the radius $r$
decreases in any fixed cone region $M_{i}\cap A_{1/2,\rho_{i}}(a_{i})$, one
could imagine the equatorial spheres $\mathbf{C}_{r}\cap\partial B_{r}$
rotating $180^{o}$, with the effect that the $M_{i}\cap\partial
B_{1/2}(a_{i})$ is glued to $M_{i}\cap\partial B_{\rho_{i}}(a_{i})$ via the
antipodal map. In this case in fact the antipodal map and the identity map are
the only two possibilities, so one could avoid this by passing to a further
subsequence.

It turns out that the second example more or less captures the behavior if all
the $\mathbf{C}_{r}$ are legitimately rotations of $\mathbf{C}$ (so
$\mathbf{C}$ is “integrable through rotations”). In this case, one can arrange
for the $\phi_{ij}|_{\partial B_{r}}$ to be small perturbations of a rotation
and dilation, and so after passing to a further subsequence we get our
required isotopies. For more general cones $\mathbf{C}$ this could fail, as
illustrated by the first example.

Instead, we can show that in fact the cones $\mathbf{C}_{r}$ cannot change at
all. In Theorem 6.3, we adapt the Lojasiewicz-Simon decay-growth estimates
[Sim83a] to prove that while the density drop is small, $M$ must stay
graphical over the same cone, _independent_ of whatever happens in smaller
regions. The intuition is that a minimal graph over a cone $\mathbf{C}$ must
either grow or decay polynomially or logarithically (in $r$) from one scale to
the next, and most importantly (as proved in [Sim83a]) if $u$ is not already
1-homogenous then there is a lower/upper bound to this growth/decay. If $u$
starts very small but then becomes too big, the graph of $u$ will start
“eating” up a definite amount of density at each scale drop. We highlight that
this decay-growth theorem requires no a priori knowledge of the singular set.
We also mention that the estimate holds in any dimension or codimension, and
for any surface with (small) $C^{2}$ mean curvature.

So, a posteriori after having constructed the cone regions by our
contradiction argument, we can use Theorem 6.3 to show that they are
legitimately graphs over a single cone. This allows our original contradiction
argument to go through. A minor modification of the above argument can also
deal with $M_{i}$ having bounded index.

So far, that deals with smooth $M_{i}$. But of course the $M_{i}$ could be
singular, and in this case it’s not really clear what $M_{i}$ being
diffeomorphic means. To adequately capture both the regular and singular
structure, we build _global_ diffeomorphisms $\phi_{ij}:B_{1/2}\to B_{1/2}$,
which take $M_{i}$ to $M_{j}$. Actually, the $\phi_{ij}$ will only be globally
bi-Lipschitz maps, and be smooth diffeomorphisms away from the singular set.
Getting better regularity of the $\phi_{ij}$ at singular points seems to be
more or less equivalent to knowing all the singular cones are integrable
through rotations.

Section 8 is devoted to constructing these maps. Multiplicity is a significant
technical thorn here. If everything were single-sheeted, the section would be
trivial (since all our maps would be $C^{1}$ perturbations of the identity),
but since we have no control over how much sheets are squished together,
gluing the various maps together is significantly more technical (since our
maps are only $C^{0}$ perturbations of $id$). The key idea is that two $C^{1}$
diffeomorphisms $\mathbb{R}\to\mathbb{R}$, which coincide with the identity
outside $[-1,1]$, are trivially isotopic just by taking an appropriate convex
combination of the two. By extension, if two $C^{1}$ diffeomorphisms act by
moving along the flowlines of some fixed, smooth vector field (having no
closed orbits), and agree outside some small set, then these can be isotoped
together along the flowlines. Even if the vector fields are different, but
very close in $C^{1}$, this will work. We then build our parameterizations
$\phi$ to have this structure.

##  4\. Preliminaries

We will mostly work in $\mathbb{R}^{8}$. If $U\subset\mathbb{R}^{8}$, then
generally all norms, lengths, areas, volumes, etc. we use in $U$ are taken
with respect to the Euclidean metric $g_{\mathrm{eucl}}$ unless otherwise
specified. Write $\overline{U}$ for the set-theoretic closure of $U$. We write
$B_{r}(a)$ for the (Euclidean) open ball in $\mathbb{R}^{8}$ centered in $a$,
and we define the open annuli
$A_{R,r}(a)=B_{R}(a)\setminus\overline{B_{r}}(a)$,
$A_{R,0}(a)=B_{R}(a)\setminus\\{a\\}$, and
$A_{\infty,r}(a)=\mathbb{R}^{8}\setminus\overline{B_{r}(a)}$.

We write $B_{r}(U)=\\{x:\mathrm{dist}(x,U)<r\\}$ for the open $r$-tubular
neighborhood of $U$, where $\mathrm{dist}(\cdot,U)$ the Euclidean distance to
$U$. Define $d_{H}(U,U^{\prime})$ to be the Hausdorff distance, i.e. the least
$r$ for which $U\subset B_{r}(U^{\prime})$ and $U^{\prime}\subset B_{r}(U)$.
Write $\mathcal{H}^{n}$ for the $n$-dimensional Hausdorff measure, and
$\omega_{n}=\mathcal{H}^{n}(B_{1}\subset\mathbb{R}^{n})$ for the volume of the
unit $n$-ball. Let $\eta_{x,r}(y)=(y-x)/r$ be the translation/dilation map.

We write $v\cdot w$ to indicate the usual Euclidean inner product, and
$|v|=(v\cdot v)^{1/2}$ for the usual Euclidean norm of a vector. If
$T(v_{1},\ldots,v_{k})$ is a mult-linear map to $\mathbb{R}^{P}$, we write
$|T|=\sup_{|v_{i}|\leq 1}|T(v_{1},\ldots,v_{k})|$ for the operator norm. If we
need to make explicit a metric $g$, we will write $|v|_{g}=g(v,v)^{1/2}$ for
the $g$-length of $v$. If $\phi:(N,g)\to(N^{\prime},g^{\prime})$ is a $C^{1}$
map between Riemannian manifolds, then for $x\in N$ we define
$|D\phi|_{x}|=\sup\\{|D\phi|_{x}(v)|_{g^{\prime}}:v\in T_{x}N,|v|_{g}\leq
1\\}$.

Given a $C^{1}$ embedded $n$-submanifold $M$ in $\mathbb{R}^{8}$, we write
$TM$, $T^{\perp}M$ for the tangent, normal bundles of
$M\subset(\mathbb{R}^{8},g_{\mathrm{eucl}})$, and for each $x\in M$ we
identity $T_{x}M$, $T_{x}^{\perp}M$ with a subspace of
$(\mathbb{R}^{8},g_{\mathrm{eucl}})$ in the obvious fashion. We write $u:M\to
M^{\perp}$ for a section of the normal bundle $T^{\perp}M$, and $\nabla$ for
the induced connection derivative on $T^{\perp}M$. Given such a $u$, we write
$\mathrm{graph}_{M}(u):=\\{x+u(x):x\in M\\}$. Define the $C^{k}$ and
$C^{k,\alpha}$ norms of $u$ w.r.t. $\nabla$. If $M_{i}$ is a sequence of
embedded submanifolds, we say $M_{i}\to M$ in $C^{k}(U)$ (with multiplicity
$m\in\mathbb{N}$) if there is a set $U^{\prime}\supset U$, and a sequence of
$C^{k}$ functions $\\{u_{ij}:M\cap U^{\prime}\to M^{\perp}\\}_{j=1}^{m}$, so
that for all $i>>1$ we have

$M_{i}\cap
U=\cup_{j=1}^{m}\mathrm{graph}_{M}(u_{ij}),\quad|u_{ij}|_{C^{k}(M\cap
U^{\prime})}\to 0\text{ as }i\to\infty.$

For $x\in M$ and $M$ as above, define the $C^{k}$ regularity scale
$r_{k}(x,M)$ to be the largest radius $r$, so that for some $n$-plane $P$ we
can find a $C^{k}$ function $v:(x+P)\cap B_{r}(x)\to P^{\perp}$ so that

$M\cap
B_{r}(x)\subset\mathrm{graph}_{x+P}(v),\quad\sum_{i=0}^{k}r^{1-i}|D^{i}v|\leq
1.$

We define the global regularity scale $r_{k}(M)=\inf_{x\in M}r_{k}(x,M)$.

Let $g$ be a $C^{2}$ metric on $U$. Write $B^{g}_{r}(x)$ for the ball of
radius $r$ centered at $x$ with respect to the metric $g$,
$\mathcal{H}^{7}_{g}$ for the Hausdorff measure w.r.t. $g$, and
$B^{g}_{r}(U)=\\{x:\mathrm{dist}_{g}(x,U)<r\\}$ for the $g$-geodesic open
$r$-tubular neighborhood around $U$. If $M$ is an embedded $C^{1}$ hypersuface
of $U$, we write $T^{\perp_{g}}M$ for the normal bundle of $M$ w.r.t. $g$, and
identity each $T_{x}^{\perp_{g}}M$ with a subspace of $\mathbb{R}^{8}$. Write
$u:M\to M^{\perp_{g}}$ for a section of normal bundle, and $\nabla^{g}$ for
the induced connection derivative on $T^{\perp_{g}}M$ induced by the Levi-
Civita connection $\nabla^{g}$ of $g$.

If $|g-g_{\mathrm{eucl}}|\leq\delta$, then for every $x\in B_{1}$ and
$r<(1-\delta)(1-|x|)$, we have

(10) $B^{g}_{(1-\delta)r}(x)\subset B_{r}(x)\subset B^{g}_{(1+\delta)r}(x).$

If $M$ is an embedded $C^{1}$ hypersurface of $U$, then we can express

(11) $\mathcal{H}^{7}_{g}(M)=\int_{M}F(g(x),T_{x}M)d\mathcal{H}^{7}(x)$

for some analytic functional $F(g,V)$ (c.f. [SS81, Remark 1]). We have
$|F(g,V)-1|\leq c|g-g_{\mathrm{eucl}}|$, and so we get

(12)
$(1-c\delta)\mathcal{H}^{7}(M)\leq\mathcal{H}^{7}_{g}(M)\leq(1+c\delta)\mathcal{H}^{7}(M).$

for some absolute constant $c$. In the special case when $M=\\{(x,u(x)):x\in
U^{\prime}\subset\mathbb{R}^{7}\\}$ for some Lipschitz function $u$ defined on
a subset of $\mathbb{R}^{7}$, then we can rewrite (11) as
$Fd\mathcal{H}^{7}=F_{0}(g(x,u(x)),Du|_{x})dx$ for $F_{0}$ an analytic
function satisfying $F(g_{\mathrm{eucl}},z)=\sqrt{1+|z|^{2}}$.

We say a $C^{2}$ embedded hypersurface $M$ is minimal in $(U,g)$ if the first
variation of (11) vanishes at $M$, i.e. for all $X\in
C^{1}_{c}(U,\mathbb{R}^{8})$, we ask

$\displaystyle 0=\delta_{g}M(X)$
$\displaystyle:=\left.\frac{d}{dt}\right|_{t=0}\mathcal{H}^{7}_{g}(\phi_{t}(M))$
(13) $\displaystyle=\int_{M}\mathrm{div}_{M,g}(X)d\mathcal{H}^{7}_{g}$

where $\phi(x,t)=x+tX$. In the last equality
$\mathrm{div}_{M,g}(X)|_{x}=\sum_{i}g(e_{i},\nabla^{g}_{e_{i}}X)$, where
$e_{i}$ is an $g$-ON-basis for $T_{x}M$ at $x$. We write
$\mathcal{M}_{7}(U,g)$ for the space of $C^{2}$ embedded minimal hypersurface
in $U$ such that $\mathrm{sing}M$ is a discrete set.

When $M=\\{(x,u(x)):x\in B_{1}\\}$ for some $C^{2}$ function
$u:B_{1}\subset\mathbb{R}^{7}\to\mathbb{R}$, then by considering the structure
of $F_{0}$, and provided $|u|_{C^{1}(B_{1})}\leq 1$ and
$|g-g_{\mathrm{eucl}}|_{C^{1}(B_{1}^{7}\times[-2,2])}\leq\delta$ for some
absolute constant $\delta$, the $u$ will satisfy a PDE of the form

$a_{ij}D^{2}_{ij}u+b_{i}D_{i}u=f_{i}D_{i}g+f_{8}D_{8}g,$

where $a_{ij},b_{i},f_{i},f_{8}$ are analytic functions of
$g(x,u(x)),Dg|_{(x,u(x))},Du|_{x}$ satisfying $1/2\leq a_{ij}\leq 2$. In
particular, by standard elliptic theory if $g\in C^{k}$ then $u\in
C^{k,\alpha}$ for every $\alpha\in(0,1)$, and for every $\theta\in(0,1)$ we
have the estimate

(14) $|u|_{C^{k,\alpha}(B_{\theta})}\leq
c(||u||_{L^{2}(B_{1})}+|Dg|_{C^{k-1}(B_{1})}),$

where
$c=c(\theta,\alpha,k,|u|_{C^{1,\alpha}(B_{1})},|Dg|_{C^{k-1}(B_{1}\times[-2,-2])})$.
Typically we will apply (14) in normal coordinates centered at some point on
the graph of $u$, so the restrictions on $g$ and $u$ are never an issue.

For $M$ minimal in $(U,g)$, we say $M$ is stable in $(U,g)$ if the second
variation of (11) is non-negative at $M$. In other words, we ask

$\delta^{2}_{g}M(X,X):=\left.\frac{d^{2}}{dt^{t}}\right|_{t=0}\mathcal{H}^{7}_{g}(\phi_{t}(M))\geq
0,\quad\forall X\in C^{1}_{c}(U,\mathbb{R}^{8}),$

where $\phi(x,t)=x+tX$. It is well known that if
$u=\pi^{\perp_{g}}_{M}(X):M\to M^{\perp_{g}}$, then
$\delta^{2}_{g}M(X,X)=Q_{M,g}(u,u)$ for $Q$ being the quadratic form in (1).
Here $\pi_{M}^{\perp_{g}}$ denotes the $g$-orthogonal projection onto
$T^{\perp_{g}}M$. Let us define $\mathrm{index}(M,U,g)$ to be the maximal
dimension of subspace $V\subset C^{1}_{c}(M\cap U,M^{\perp_{g}})$ on which
$Q_{M,g}$ is strictly negative definite, i.e. so that $Q_{M,g}(\phi,\phi)<0$
for all $\phi\in V$. Observe the trivial inclusion
$\mathrm{index}(M,U^{\prime},g)\leq\mathrm{index}(M,U,g)$ if
$U^{\prime}\subset U$.

We note that by [Wic14, Section 18], if $\mathrm{index}(M,U,g)=0$ and
$\mathcal{H}^{6}(U\cap\overline{M}\setminus M)=0$, then
$U\cap\overline{M}\setminus M$ consists of isolated points, and hence by an
elementary cutoff function argument $M$ is stable in $(U,g)$. This motivates
our definition of $\mathcal{M}_{7}$.

If $M$ is closed (as sets) in $U$, then $Q_{M,g}$ can be written

$Q_{M,g}(u,u)=-\int_{M}g(u,\Delta_{g}u+|A_{M,g}|^{2}u+Ric_{g}(u,\cdot))d\mathcal{H}^{7}_{g}=:-\int_{M}g(u,L_{M,g}u)d\mathcal{H}^{7}_{g},$

where $\Delta_{g}$ is the connection Laplacian on $T^{\perp_{g}}M$. $L_{M,g}$
is a self-adjoint elliptic operator on $C^{2}_{c}(M,M^{\perp_{g}})$. In this
case, for every $U^{\prime}\subset\subset U$ with $C^{1}$ boundary
intersecting $M$ transversely, $\mathrm{index}(M,U^{\prime},g)$ is finite and
consists of the number of negative eigenvalues for $L_{M,g}$ under Dirichlet
zero boundary conditions on $M\cap U^{\prime}$.

As discussed in the Introduction, singular minimal surfaces with finite index,
in the sense that we’ve described above, arise naturally in various min-max
constructions (see e.g. [Li20], [Hie18], [Gas20]). We note that if $M$ is a
minimal cone in $(\mathbb{R}^{8},g_{\mathrm{eucl}})$ having
$\mathcal{H}^{6}(\overline{M}\setminus M)=0$, it’s easy to check that either
$M$ is stable, or $\mathrm{index}(M,\mathbb{R}^{8},g_{\mathrm{eucl}})=\infty$.
So any kind of finite index must be “macroscopic.” This can be made more
precise below.

For any $x\in U$, we define the “stability radius” $r_{s}(x)\equiv
r_{s}(x,M,U,g)$ to be the largest radius $r_{s}$ with the property that
$B_{r_{s}}(x)\subset U$, and $\mathrm{index}(M,B_{r_{s}}(x),g)=0$. Essentially
the same quantity, as well as the below Lemma 4.1, appeared in [Son20]. Since
it is very short we include a proof here for the readers’ convenience. We
remark that [Wan20, Corollary 3.8] has shown a converse to Lemma 4.1, i.e. if
$M\in\mathcal{M}_{7}(U,g)$ and $r_{s}>0$ everywhere in $U$ then $M$ has
bounded index on compact subsets.

######  Lemma 4.1.

Let $M$ be a $C^{2}$ embedded minimal hypersurface in $(U,g)$, for
$U\subset\mathbb{R}^{8}$ open and $g$ a $C^{2}$ metric on $U$. Suppose
$\mathrm{index}(M,U,g)<\infty$ and $\mathcal{H}^{6}(U\cap\overline{M}\setminus
M)=0$. Then $r_{s}>0$ on $U$, and (hence) $M\in\mathcal{M}_{7}(U,g)$.

###### Proof.

If $x\in M$, then this follows trivially by the Poincare inequality in
$B_{r}(x)\cap M$, for $r$ small. Suppose $x\in\overline{M}\setminus M$ but
$r_{s}(x)=0$. Then by our hypothesis and [Wic14] (as in our discussion above),
we must have $\mathrm{index}(M,B_{r}(x),g)>0$ for every $r>0$ small. We can
therefore find a $C^{1}_{c}$ functions $u_{r}:M\cap B_{r}(x)\to M^{\perp_{g}}$
with $Q_{M,g}(u,u)<0$ and $\mathrm{spt}u_{r}\cap\overline{M}\setminus
M=\emptyset$. But then for any such $r$ we can find an $r^{\prime}>0$ with
$\mathrm{spt}u_{r}\cap B_{r^{\prime}}(x)=\emptyset$, and thereby construct an
infinite-dimensional subspace $V\subset C^{1}_{c}(M,M^{\perp_{g}})$ on which
$Q_{M,g}$ is strictly negative definite. This is a contradiction, so we must
have $r_{s}(x)>0$. ∎

It will be convenient to use the notion of varifold. Recall that if
$U\subset\mathbb{R}^{8}$, then an integral $7$-varifold $V$ in $(U,g)$ is a
Radon measure on the Grassmanian bundle $G_{7}(TU)\equiv
U\times\mathrm{Gr}(7,8)$ that takes the form

(15)
$V(\phi(x,S))=\int_{M_{V}}\phi(x,T_{x}M_{V})\theta_{V}(x)d\mathcal{H}^{7}_{g}(x)$

for some countably $7$-rectifiable set $M_{V}$, and some positive, integer-
valued $\mathcal{H}^{7}_{g}$-measurable function $\theta_{V}$. The mass
measure $\mu_{V}=\mathcal{H}^{7}_{g}\llcorner\theta_{V}\llcorner M_{V}$ is the
pushforward of $V$ under the projection map $G_{7}(TU)\to U$. We write
$\mathrm{spt}V=\mathrm{spt}\mu_{V}$, and define $\mathrm{reg}V$ to be the set
of points $x\in\mathrm{spt}V$ having the property that for some $r>0$,
$\mathrm{spt}V\cap B_{r}(x)$ is a closed (as sets) embedded hypersurface of
$B_{r}(x)$. We define the closed set
$\mathrm{sing}V=\mathrm{spt}V\setminus\mathrm{reg}V$. If $V_{i}$, $V$ are
$7$-varifolds in $\mathbb{R}^{8}$ and $V_{i}\to V$ as varifolds, we say
$V_{i}\to V$ in $C^{k}(U)$ if for some $U^{\prime}\supset U$ and for $i>>1$,
the $\mathrm{spt}V_{i}\cap U^{\prime}$ and $\mathrm{spt}V\cap U^{\prime}$ are
$C^{k}$ embedded hypersurfaces, and $\mathrm{spt}V_{i}\to\mathrm{spt}V$ in
$C^{k}(U)$.

If $M$ is a $C^{1}$ hypersurface of $U$, then $M$ induces a natural varifold
$[M]_{g}$ by taking in (15) $M_{V}=M$ and $\theta_{V}=1$. Write
$[M]\equiv[M]_{g_{\mathrm{eucl}}}$. Given any $C^{1}$, proper map
$f:(U,g)\to(U^{\prime},g^{\prime})$, we define the pushforward $f_{\sharp}V$
to be the integral varifold $f_{\sharp}V$ defined by

$(f_{\sharp}V)(\phi(x,S))=\int\phi(f(x),Df|_{x}S)J_{S}f(x)dV(x,S),$

where, given any $g|_{x}$-ON basis $e_{i}$ for $S$,
$J_{S}(x)=\det(g^{\prime}|_{f(x)}(Df|_{x}(e_{i}),Df|_{x}(e_{j}))^{1/2}$.
Equivalently, by the area formula, in the notation of (15)
$M_{f_{\sharp}V}=f(M_{V})$ and
$\theta_{f_{\sharp}V}(x)=\sum_{y\in\theta^{-1}(x)}\theta_{V}(y)$. Similarly,
$f_{\sharp}[M]_{g}=[f(M)]_{g^{\prime}}$.

$V$ is stationary in $(U,g)$ if the first variation $\delta_{g}V$ vanishes,
i.e. if for all $U^{\prime}\subset\subset U$ and $X\in
C^{1}_{c}(U^{\prime},\mathbb{R}^{8})$, we have

(16)
$0=\delta_{g}V(X):=\left.\frac{d}{dt}\right|_{t=0}\mu_{(Id+tX)_{\sharp}V}(U^{\prime})=\int_{U^{\prime}}\mathrm{div}_{S,g}(X)dV(x,S),$

where $\mathrm{div}_{S,g}(X)=\sum_{i}g(e_{i},\nabla^{g}_{e_{i}}X)$ for any
$g|_{x}$-ON basis of $S$. If $M$ is a minimal surface in $(U,g)$, then
$[M]_{g}$ is stationary in $(U,g)$.

Let $V$ be a stationary integral varifold in $(B_{1},g)$, and suppose $g$ is a
$C^{2}$ metric satisfying $|g-g_{\mathrm{eucl}}|+|Dg|\leq\Gamma|x|$. Then by
(15), (16) we have the inequality

$\displaystyle|\delta_{g_{\mathrm{eucl}}}V(X)|$
$\displaystyle=\left|\int\mathrm{div}_{T_{x}M_{V},g_{\mathrm{eucl}}}(X)d\mu_{V}\right|$
$\displaystyle\leq c\Gamma\int|x||DX|+|X|d\mu_{V}(x)\quad\forall X\in
C^{1}_{c}(B_{1},\mathbb{R}^{8}),$

for $c$ an absolute constant. Therefore by plugging in $X=\phi(|x|/\rho)x$ for
$\phi$ being (a suitable smooth approximation for) $1_{(-\infty,1]}$, we
obtain the monotonicity

(17)
$\int_{A_{r,s}}\frac{|\pi^{\perp}_{V}(x)|^{2}}{|x|^{7+2}}d\mu_{V}(x)\leq(1+c\Gamma
r)^{7+1}\theta_{V}(0,r)-(1+c\Gamma s)^{7+1}\theta_{V}(0,s).$

for all $0<s\leq r\leq 1$, for $c$ an absolute constant, and for
$\theta_{V}(x,r)=\omega_{7}^{-1}r^{-7}\mu_{V}(B_{r}(x))$ the density ratio of
$V$. Here $\pi^{\perp}_{V}(x)$ is the _Euclidean_ projection of the position
vector $x$ onto the _Euclidean_ normal direction $T^{\perp}_{x}M_{V}$. Of
course if $g=g_{\mathrm{eucl}}$, then (17) reduces to the usual sharp
monotonicity formula. We note also that, since $X$ is smooth near $0$ and
$X(0)=0$, (17) continues to hold if $V$ is only stationary in
$(B_{1}\setminus\\{0\\},g)$, provided $\mu_{V}(B_{1}\setminus\\{0\\})<\infty$,
in which case $V$ extends to a stationary integral varifold in $(B_{1},g)$.

Suppose now only $|g-g_{\mathrm{eucl}}|_{C^{2}(B_{1})}\leq\delta$. For each
$x\in B_{1}$ and $r<(1-\delta)(1-|x|)$ we can find a $C^{2}$ coordinte
transformation $z=\phi_{x}(y)$ in $B_{r}(x)$, so that the $(z^{i})$ form a
choice of normal coordinates of $g$ at $x$. Applying the previous paragraph in
this coordinate chart, and using (10), (12), we obtain the monotonicity

(18) $\theta_{V}(x,s)\leq(1+c_{0}\delta r)\theta_{V}(x,r)\quad\forall x\in
B_{1},0<s<r<(1-\delta)r,$

where $c_{0}$ is an absolute constant. (18) implies that
$\theta_{V}(x):=\lim_{r\to 0}\theta_{V}(x,r)$ exists, is upper-semi-
continuous, and (since $V$ is integral) $\geq 1$ on $\mathrm{spt}V$. Moreover,
given any $\sigma<1$ and $x\in\mathrm{spt}V\cap B_{\sigma}$, we have the
Ahlfors regularity

(19)
$\frac{1}{2}\leq\theta_{V}(x,r)\leq\frac{1+c_{0}}{(1-\sigma)^{n}}\mu_{V}(B_{1}),\quad
0<r<\min\\{1-|x|,\frac{1}{c_{0}\delta}\\}.$

We define $\mathcal{ISV}_{7}(U,g)$ to be the class of stationary integral
$7$-varifolds $V$ in $(U,g)$ with the property that $\mathrm{sing}V$ is a
discrete set of points. Every such $V$ satisfies
$\mathrm{reg}V\in\mathcal{M}_{7}(U,g)$, and more generally has the following
structure.

######  Lemma 4.2.

A varifold $V\in\mathcal{ISV}_{7}(U,g)$ if and only if for every
$U^{\prime}\subset\subset U$ there is a $k\in\mathbb{N}$ so that we can write

(20) $V\llcorner U^{\prime}=\sum_{i=1}^{k}m_{i}[M_{i}]_{g}$

for $m_{i}\in\mathbb{N}$, $M_{i}\in\mathcal{M}_{7}(U^{\prime},g)$, and with
$\overline{M_{i}}$ being disjoint.

###### Proof.

Let us write $\mathrm{reg}V\cap U^{\prime}=\cup_{i}M_{i}$ for each $M_{i}$
being a connected, disjoint, $C^{2}$, embedded hypersurface of $U^{\prime}$.
Since $\cup_{i}(\overline{M_{i}}\setminus
M_{i})\subset\overline{\mathrm{reg}V}\setminus\mathrm{reg}V=\mathrm{sing}V$,
each $\overline{M_{i}}\setminus M_{i}$ consists of a discrete set of points.
By the constancy theorem ([Sim83b, Chapter 8, Theorem 4.1]) $\theta_{V}$ is
locally constant on each $M_{i}$, and hence
$\theta_{V}|_{M_{i}}=m_{i}\in\mathbb{N}$. By stationarity we get that each
$M_{i}$ is minimal in $(U^{\prime},g)$, and hence
$M_{i}\in\mathcal{M}_{7}(U^{\prime},g)$. By applying (19) to each stationary
varifold $[M_{i}]_{g}$, we get that the collection $\\{M_{i}\\}_{i}$ is
finite, and by the maximum principle of [Ilm96], the
$\\{\overline{M_{i}}\\}_{i}$ are disjoint. This proves the “only if” part of
the Lemma; the “if” part is trivial. ∎

######  Remark 4.3 (A note on scaling).

Let $B_{R}(a)\subset\mathbb{R}^{8}$ and $g$ be a $C^{2}$ metric on $B_{R}(a)$.
If $V\in\mathcal{ISV}_{7}(B_{R}(a),g)$, and we define
$g^{\prime}=g\circ\eta_{a,R}^{-1}$ and $V^{\prime}=(\eta_{a,R})_{\sharp}V$
where we view $\eta_{a,R}$ as a map $(B_{R}(a),g)\to(B_{1}(0),g^{\prime})$
(i.e. so _not_ an isometry), then
$V^{\prime}\in\mathcal{ISV}_{7}(B_{1}(0),g^{\prime})$, and
$\mu_{V^{\prime}}(U)=R^{-n}\mu_{V}(\eta_{a,R}^{-1}(U))$ for all $U\subset
B_{1}$. We shall always interpret the rescaling $(\eta_{a,R})_{\sharp}V$ in
this fashion. Of course if $M\in\mathcal{M}_{7}(B_{R}(a),g)$, then
$(\eta_{a,R})_{\sharp}[M]_{g}=[\eta_{a,R}(M)]_{g^{\prime}}$.

There is good compactness theory for stationary integral varifolds, but for
our purposes we will make use of the following compactness theory for
$\mathcal{ISV}_{7}$. Given $V\in\mathcal{ISV}_{7}(U,g)$ we define
$\mathrm{index}(V,U,g):=\mathrm{index}(\mathrm{reg}V,U,g)$.

######  Lemma 4.4.

Let $g_{i},g$ be $C^{2}$ metrics on $U$, such that $g_{i}\to g$ in $C^{2}(U)$.
Let $V_{i}\in\mathcal{ISV}_{7}(U,g_{i})$ be a sequence satisfying

(21) $\sup_{i}\mu_{V_{i}}(U)<\infty,\quad\mathrm{index}(V_{i},U,g)\leq
I<\infty$

for some $I\in\mathbb{N}$.

Then after passing to a subsequence, we can find a
$V\in\mathcal{ISV}_{7}(U,g)$, and at most $I$ points
$\mathcal{I}\subset\mathrm{spt}V\cap U$, so that: $V_{i}\to V$ as varifolds in
$U$; $V_{i}\to V$ in $C^{2}$ on compact subsets of
$U\setminus(\mathrm{sing}V\cup\mathcal{I})$; $\mathrm{index}(V,U,g)\leq I$;
and $\mathrm{index}(V_{i},B_{r}(y),g_{i})\geq 1$ for every $r>0$,
$y\in\mathcal{I}$, $i>>1$. Additionally, for any collection of disjoint balls
$\\{B_{r_{\alpha}}(x_{\alpha})\\}_{\alpha}\subset U$, we have

(22)
$\displaystyle\sum_{\alpha}\mathrm{index}(V_{i},B_{r_{\alpha}}(x_{\alpha}),g_{i})\leq
I\quad\forall i.$

###### Proof.

Let $\mathcal{I}$ consist of the points $y\in U$ with the property that
$\limsup_{i}r_{s}(y,V_{i})=0$. Trivially we have
$\mathrm{index}(V_{i},B_{r}(y),g_{i})\geq 1$ for every $r>0$,
$y\in\mathcal{I}$, $i>>1$. We first show $\\#\mathcal{I}\leq I$. Suppose, on
the contrary, we can find $\\{y_{1},\ldots,y_{I+1}\\}\subset\mathcal{I}$. Then
for $i>>1$, we can find radii $r_{i}$ so that
$\\{B_{r_{i}}(y_{j})\\}_{j}\subset B_{1}$ are disjoint, and $C^{1}_{c}$
functions $u_{ij}:\mathrm{reg}V_{i}\cap
B_{r_{i}}(y_{j})\to\mathrm{reg}V_{i}^{\perp_{g_{i}}}$ so that
$Q_{\mathrm{reg}V_{i},g_{i}}(u_{ij},u_{ij})<0$. These functions are trivially
linearly-independent, and form a strictly-negative-definite subspace for
$Q_{\mathrm{reg}V_{i},g_{i}}$, and hence $\mathrm{index}(V_{i},U,g_{i})\geq
I+1$. This is a contradiction. An essentially verbatim argument shows (22).

Take any $x\in B_{1}\setminus\mathcal{I}$. There is a radius $r=r_{x}>0$ so
that, after passing to a subsequence, $B_{2r}(x)\subset U$ and every $V_{i}$
is stable in $(B_{r}(x),g_{i})$. Each $\mathrm{reg}V_{i}\cap B_{r}(x)$ is
orientable (being a set-theoretically-closed embedded hypersurface in the
simply-connected region $B_{r}(x)\setminus\mathrm{sing}V_{i}$), and so by
[SS81, Theorems 1 and 2], (11), and $C^{0}$ convergence $g_{i}\to g$, after
passing to a subsequence we can find a $V\in\mathcal{ISV}_{7}(B_{r}(x),g)$ so
that $[\mathrm{spt}V_{i}\cap B_{r}(x)]_{g_{i}}\to V$ as varifolds in
$B_{r}(x)$, and $\mathrm{spt}V_{i}\to\mathrm{spt}V$ in $C^{1,\alpha}$ on
compact subsets of $B_{r}(x)\setminus\mathrm{sing}V$ for every
$\alpha\in(0,1)$. Passing to a further subsequence, (14) and our $C^{2}$
convergence $g_{i}\to g$ imply that $\mathrm{spt}V_{i}\to\mathrm{spt}V$ in
$C^{2}$ on compact subsets of $B_{r}(x)\setminus\mathrm{sing}V$. By (20) and
(19), we can pass to yet a further subsequence, and modify the multiplicities
of $V$, so that $V_{i}\to V$ as varifolds in $B_{r}(x)$.

The previous paragraph implies that there is an
$V_{x}\in\mathcal{ISV}_{7}(B_{r}(x),g)$ so that $V_{i}\to V_{x}$ as varifolds
in $B_{r}(x)$, and $V_{i}\to V_{x}$ in $C^{2}$ on compact subsets of
$B_{r}(x)\setminus\mathrm{sing}V_{x}$. Now by considering a countable subcover
of $\\{B_{r_{x}}(x)\\}_{x\in U\setminus\mathcal{I}}$, and applying a
diagonalization argument, we can obtain an
$V\in\mathcal{ISV}_{7}(U\setminus\mathcal{I},g)$ so that $V_{i}\to V$ as
varifolds in $U\setminus\mathcal{I}$, and in $C^{2}$ on compact subsets of
$U\setminus(\mathrm{sing}V\cup\mathcal{I})$. Lower-semi-continuity of mass
implies $\mu_{V}(U\setminus\mathcal{I})<\infty$, so by (17) $V$ extends to a
stationary integral varifold in $(U,g)$, and $V_{i}\to V$ as varifolds in $U$.

We show $\mathrm{index}(V,U,g)\leq I$, which will by Lemma 4.1 imply
$V\in\mathcal{ISV}_{7}(U,g)$. Suppose, otherwise: there is a linear subspace
$V\subset C^{1}_{c}(\mathrm{reg}V,\mathrm{reg}V^{\perp_{g}})$ of dimension
$I+1$, so that $Q_{\mathrm{reg}V,g}(u,u)<0$ for all $u\in V$. Let
$u_{1},\ldots,u_{I+1}$ be a basis for $V$. Then there is a $q<0$ so that

(23)
$Q_{\mathrm{reg}V,g}(\sum_{j}\lambda_{j}u_{j},\sum_{j}\lambda_{j}u_{j})\leq
q<0$

for all collections $\lambda_{j}$ satisfying $\sum_{j}\lambda_{j}^{2}=1$.

Take $\eta:\mathbb{R}\to\mathbb{R}$ be a fixed, smooth, increasing function
satisfying

$\eta|_{(-\infty,1/4]}\equiv 0,\quad\eta|_{[3/4,\infty)}\equiv
1,\quad|\eta^{\prime}|\leq 10,$

and then for $\sigma>0$ define
$\phi_{\sigma}(x)=\eta(\sigma^{-1}\mathrm{dist}(x,\mathcal{I}))$, and
$u_{j,\sigma}=\phi_{\sigma}u_{j}$. Given any $U^{\prime}\subset\subset
U\setminus\mathcal{I}$, then $u_{i,\sigma}=u_{i}$ on $U^{\prime}$ for $\sigma$
sufficiently small. Since $\mathcal{I}$ is discrete (codimension $7$), (19)
implies $Q_{\mathrm{reg}V,g}(u_{j,\sigma},u_{k,\sigma})\to
Q_{\mathrm{reg}V,g}(u_{j},u_{k})$ as $\sigma\to 0$, for any $j,k$. We deduce
(after taking $\sigma$ sufficiently small, and replacing $q$ in (23) with
$q/2$), that it suffices to assume that the $u_{i}$ are supported away from
$\mathcal{I}$.

Now $C^{2}$ convergence $V_{i}\to V$ implies that we can find open sets
$\mathrm{spt}u_{j}\subset W_{j}\subset
U\setminus(\mathcal{I}\cup\mathrm{sing}V)$, and $C^{2}$ functions
$w_{ij}:W_{j}\cap\mathrm{reg}V\to\mathrm{reg}V^{\perp_{g}}$ such that

$\mathrm{reg}V_{i}\cap W_{j}\supset\mathrm{graph}_{\mathrm{reg}V}(w_{ij})\cap
W_{j}=:G_{ij}$

and $|w_{ij}|_{C^{2}}\to 0$ as $i\to\infty$. Define $u_{ij}\in
C^{1}_{c}(\mathrm{reg}V_{i},V_{i}^{\perp_{g_{i}}})$ by setting
$u_{ij}(y=x+w_{ij}(x))=\pi_{G_{ij}}^{\perp_{g_{i}}}(u_{j}(x))$ for $y\in
G_{ij}$, and $u_{ij}=0$ else. Here $\pi^{\perp_{g_{i}}}_{G_{ij}}|_{y}$ denotes
the $g_{i}$-orthogonal projection to the normal space
$T_{y}^{\perp_{g_{i}}}G_{ij}$.

Since $\mathrm{index}(V_{i},U,g)\leq I$, and the $u_{ij}$ are linearly
independent for $i>>1$, we can find numbers $\lambda_{ij}$ so that
$Q_{\mathrm{reg}V_{i},g_{i}}(\sum_{j}\lambda_{ij}u_{ij},\sum_{j}\lambda_{ij}u_{j})\geq
0$, and (WLOG) $\sum_{j}\lambda_{ij}^{2}=1$. Passing to a subsequence, we get
$\lambda_{ij}\to\lambda_{j}$ for each $j$, and hence by $C^{2}$ convergence
$V_{i}\to V$, $g_{i}\to g$, we get

$Q_{\mathrm{reg}V_{i},g_{i}}(\sum_{j}\lambda_{ij}u_{ij},\sum_{j}\lambda_{ij}u_{ij})\to
Q_{\mathrm{reg}V,g}(\sum_{j}\lambda_{j}u_{j},\sum_{j}\lambda_{j}u_{j})\geq 0.$

However this contradictions (23). We must therefore have
$\mathrm{index}(V,U,g)\leq I$. ∎

##  5\. Stable cones

In this section we work in $\mathbb{R}^{8}$. We let $\mathcal{C}$ be the
collection of stable minimal hypercones $\mathbf{C}^{7}$ in $\mathbb{R}^{8}$
which are smooth, closed (as sets), embedded hypersurfaces away from $0$. By
[SS81], [Wic14], if $\mathbf{C}\subset\mathbb{R}^{8}$ is a smooth, stable,
embedded hypersurface and
$\mathcal{H}^{6}(\overline{\mathbf{C}}\setminus\mathbf{C})=0$, then
$\mathbf{C}\in\mathcal{C}$. Note that each link $\mathbf{C}\cap\partial B_{1}$
is necessarily orientable (as a closed, embedded, codimension-one submanifold
of a simply-connected manifold) and connected (by the maximum principle, or
more generally by Frankel’s theorem).

Given $\Lambda>0$, let $\mathcal{C}_{\Lambda}\subset\mathcal{C}$ denote the
collection of $\mathbf{C}\in\mathcal{C}$ with
$\theta_{\mathbf{C}}(0)\leq\Lambda$. Given $\mathbf{C}\in\mathcal{C}$, define
$\mathcal{C}(\mathbf{C})$ to be the collection of
$\mathbf{C}^{\prime}\in\mathcal{C}$ with property that
$\theta_{\mathbf{C}^{\prime}}(0)=\theta_{\mathbf{C}}(0)$, and there is a
$C^{2}$ diffeomorphism $\phi:\partial B_{1}\to\partial B_{1}$ such that
$\phi(\mathbf{C}\cap\partial B_{1})=\mathbf{C}^{\prime}\cap\partial B_{1}$. Of
course the condition $\mathbf{C}^{\prime}\in\mathcal{C}(\mathbf{C})$ is both
symmetric and transitive.

Given $\mathbf{C}\in\mathcal{C}$ and function $u:\mathbf{C}\cap
A_{r,s}(0)\to\mathbf{C}^{\perp}$, let us define the conical graph

$G_{\mathbf{C}}(u)\cap
A_{r,s}(0)=\left\\{\frac{x+u(x)}{|x+u(x)|}|x|:x\in\mathbf{C}\cap
A_{r,s}(0)\right\\}.$

If $a\in\mathbb{R}^{8}$ and $u$ is defined instead on $(a+\mathbf{C})\cap
A_{r,s}(a)$, we define

$G_{a+\mathbf{C}}(u)\cap A_{r,s}(a)=a+G_{\mathbf{C}}(u(a+\cdot))\cap
A_{r,s}(0).$

When $|u|_{C^{1}}$ is small this notion of graph is effectively equivalent to
the “usual” graph $\mathrm{graph}_{a+\mathbf{C}}(u)$, but is more convenient
to work with in the conical setting. Similarly, if $\Sigma$ is a $C^{2}$
closed embedded hypersurface in $\partial B_{1}$, and
$v:\Sigma\to\Sigma^{\perp}$, then let us write

$G_{\Sigma}(v)=\left\\{\frac{\theta+v(\theta)}{\sqrt{1+|v|^{2}}}:\theta\in\Sigma\right\\}$

for the spherical graph

The key facts we will use about $\mathcal{C}$ are listed below.

######  Theorem 5.1.

The following are true.

  1. (1)

Given any sequence $\mathbf{C}_{i}\in\mathcal{C}$ with
$\sup_{i}\theta_{\mathbf{C}_{i}}(0)<\infty$, there is a subsequence
$i^{\prime}$ and cone $\mathbf{C}\in\mathcal{C}$ so that
$\mathbf{C}_{i^{\prime}}\cap\partial B_{1}$ converges to
$\mathbf{C}\cap\partial B_{1}$ smoothly with multiplicity-one, and moreover so
that $\theta_{\mathbf{C}_{i^{\prime}}}(0)=\theta_{\mathbf{C}}(0)$ for all
$i^{\prime}$.

  2. (2)

The set of densities $\\{\theta_{\mathbf{C}}(0):\mathbf{C}\in\mathcal{C}\\}$
forms a discrete set $1=\theta_{0}<\theta_{1}<\theta_{2}<\ldots$.

  3. (3)

Given $\Lambda>0$, there is an $\epsilon_{0}(\Lambda)>0$ so that if
$\mathbf{C},\mathbf{C}^{\prime}\in\mathcal{C}_{\Lambda}$ satisfy

$d_{H}(\mathbf{C}\cap\partial B_{1},\mathbf{C}^{\prime}\cap\partial
B_{1})\leq\epsilon\leq\epsilon_{0},$

then $\mathbf{C}^{\prime}\in\mathcal{C}(\mathbf{C})$, and in fact we can write

$\mathbf{C}^{\prime}=G_{\mathbf{C}}(v),\quad|x|^{1-k}|\nabla^{k}v|\leq
c(\Lambda,k)\epsilon$

for a smooth, $1$-homogenous function
$v:\mathbf{C}\setminus\\{0\\}\to\mathbf{C}^{\perp}$.

  4. (4)

Additionally, for $\mathbf{C}\in\mathcal{C}_{\Lambda}$ we have
$r_{3}(\mathbf{C}\cap\partial B_{1})\geq\epsilon_{0}$.

###### Proof.

The main ingredient is the Lojasiewicz-Simon inequality of [Sim83a]. Let
$\Sigma$ be any smooth, closed, embedded minimal submanifold of $S^{7}$. Then
[Sim83a, Theorem 3] implies there is a $\sigma(\Sigma,\mu)>0$ and
$\theta(\Sigma)\in(0,1/2)$ so that given any $C^{2,\mu}$ function
$u:\Sigma\to\Sigma^{\perp}$ (for $\mu\in(0,1)$) with
$|u|_{C^{2,\mu}(\Sigma)}\leq\sigma$ we have the inequality

$|\mathcal{H}^{6}(G_{\Sigma}(u))-\mathcal{H}^{6}(G_{\Sigma}(0))|^{1-\theta}\leq||\mathcal{M}(u)||_{L^{2}(\Sigma)},$

where $\mathcal{M}(u)$ is the negative $L^{2}(\Sigma)$-gradient of
$u\mapsto\mathcal{H}^{6}(G_{\Sigma}(u))$ (see Appendix 11). In particular,
$G_{\Sigma}(u)$ is minimal in $S^{7}$ precisely when $\mathcal{M}(u)=0$, in
which case we have $\mathcal{H}^{6}(G_{\Sigma}(u))=\mathcal{H}^{6}(\Sigma)$.
Of course minimality of $\Sigma$ in $S^{7}$ is equivalent to minimality of the
cone over $\Sigma$ in $\mathbb{R}^{8}$.

The secondary ingredient is smooth, multiplicity-one compactness for
$\mathcal{C}_{\Lambda}$. Let us first note that every smooth, closed, minimal
$6$-submanifold in $S^{7}$ is necessarily connected and orientable. Given a
sequence $\mathbf{C}_{i}$ as in the Theorem, then by the usual varifold
compactness, sheeting, and dimension reducing argument (e.g. [SS81, Theorem 2]
or Lemma 4.4) we can find a $\mathbf{C}\in\mathcal{C}$ and $m\in\mathbb{N}$ so
that $\mathbf{C}_{i}\to\mathbf{C}$ smoothly with multiplicity $m$ away from
$0$. However since each $\mathbf{C}_{i}\cap\partial B_{1}$,
$\mathbf{C}\cap\partial B_{1}$ is connected while $\partial B_{1}=S^{7}$ is
simply-connected, we must have $m=1$. Theorem 5.1 follows in an obvious way
from these two ingredients. ∎

We require some auxilary Lemmas. Our first shows how the sequence compactness
of Theorem 5.1 implies a topological compactness as well.

######  Lemma 5.2.

Take $\Lambda>0$, and a mapping
$\mathbf{C}\in\mathcal{C}_{\Lambda}\mapsto\delta_{\mathbf{C}}\in(0,\infty)$
assigning to every cone in $\mathcal{C}_{\Lambda}$ a positive “radius.” Then
we can find a finite collection
$\mathbf{C}_{1},\ldots,\mathbf{C}_{N}\subset\mathcal{C}_{\Lambda}$, so that
for every $\mathbf{C}\in\mathcal{C}_{\Lambda}$ we have

$d_{H}(\mathbf{C}\cap\partial B_{1},\mathbf{C}_{i}\cap\partial
B_{1})<\delta_{\mathbf{C}_{i}}$

for some $i\in\\{1,\ldots,N\\}$.

###### Proof.

Let $\mathcal{V}$ be the space of stationary integral $6$-varifolds in $S^{7}$
(stationary with respect to the spherical metric), having total mass $\leq
2\Lambda$, and let this space have the topology induced by varifold
convergence. The usual compactness theorem for stationary varifolds [Sim83b,
Chapter 8, Theorem 5.8] implies $\mathcal{V}$ is sequentially compact, and if
we include $\mathcal{C}_{\Lambda}$ into $\mathcal{V}$ by identifying
$\mathbf{C}$ with the varifold $[\mathbf{C}\cap\partial B_{1}]$, then by
Theorem 5.1 $\mathcal{C}_{\Lambda}$ is closed in $\mathcal{V}$.

Define the metric $d$ on $\mathcal{V}$ as follows: fix a countably-dense
subcollection $\\{\phi_{i}\\}_{i}\subset\\{\phi\in
C^{0}(B_{2}):|\phi|_{C^{0}(B_{2})}\leq 1\\}$, and then given
$V,W\in\mathcal{V}$ set

(24) $d(V,W)=\sum_{i}2^{-i}\left|\int\phi d\mu_{V}-\int\phi d\mu_{W}\right|.$

Since any $V\in\mathcal{V}$ is uniquely determined by its mass measure
$\mu_{V}$, since $\mathcal{V}$ is sequentially compact, and since every
$V\in\mathcal{V}$ has $\mu_{V}(S^{7})\leq 2\Lambda$, it’s straightfoward to
verify that the varifold topology on $\mathcal{V}$ coincides with the topology
induced by $d$. Moreover, the monotonicity formula and sequential compactness
imply: given any $\epsilon>0$, there is a $\delta>0$ so that for any
$V,W\in\mathcal{V}$:

(25) $d(V,W)<\delta\quad\implies\quad
d_{H}(\mathrm{spt}V,\mathrm{spt}W)<\epsilon.$

A converse relation holds if $V,W\in\mathcal{C}_{\Lambda}\subset\mathcal{V}$.

By (25) we can find an $\epsilon_{\mathbf{C}}>0$ for each $\mathbf{C}$ so that

(26) $\displaystyle U_{\mathbf{C}}$
$\displaystyle:=\\{\mathbf{C}^{\prime}\in\mathcal{C}_{\Lambda}:d([\mathbf{C}\cap\partial
B_{1}],[\mathbf{C}^{\prime}\cap\partial B_{1}])<\epsilon_{\mathbf{C}}\\}$ (27)
$\displaystyle\subset\\{\mathbf{C}^{\prime}\in\mathcal{C}_{\Lambda}:d_{H}(\mathbf{C}\cap\partial
B_{1},\mathbf{C}^{\prime}\cap\partial B_{1})<\delta_{\mathbf{C}}\\}$

(of course $\mathrm{spt}[\mathbf{C}\cap\partial B_{1}]=\mathbf{C}\cap\partial
B_{1}$ since $\mathbf{C}$ is smooth away from $0$). So the $U_{\mathbf{C}}$
form an open cover of $\mathcal{C}_{\Lambda}\subset\mathcal{V}$. Since
compactness and sequential compactness are equivalent for a metric space, and
$\mathbf{C}_{\Lambda}$ is closed in $\mathcal{V}$, we can find a finite
subcollection $U_{\mathbf{C}_{1}},\ldots,U_{\mathbf{C}_{N}}$ which covers
$\mathbf{C}_{\Lambda}$. The Lemma then follows by (25). ∎

The second is a quantification of the fact that none of the cones in
$\mathbf{C}$ have any translational symmetry.

######  Lemma 5.3.

Given $\epsilon>0$, $\Lambda>0$ there is a $\beta(\Lambda,\epsilon)$ so that
if $\mathbf{C},\mathbf{C}^{\prime}\in\mathcal{C}_{\Lambda}$, and
$a\in\mathbb{R}^{8}$ satisfy

$d_{H}((a+\mathbf{C}^{\prime})\cap A_{1,1/2},\mathbf{C}\cap
A_{1,1/2})\leq\beta,$

then $|a|\leq\epsilon$, $d_{H}(\mathbf{C}^{\prime}\cap B_{1},\mathbf{C}\cap
B_{1})\leq\epsilon$, and
$\theta_{\mathbf{C}}(0)=\theta_{\mathbf{C}^{\prime}}(0)$. If $\mathbf{C}$ is
planar, then we require $a\in{\mathbf{C}^{\prime}}^{\perp}$.

###### Proof.

Suppose, towards a a contradiction, the lemma failed: there was a sequence
$\mathbf{C}_{i},\mathbf{C}_{i}^{\prime}\in\mathcal{C}_{l}$,
$a_{i}\in\mathbb{R}^{8}$, so that

(28) $d_{H}((a_{i}+\mathbf{C}_{i}^{\prime})\cap A_{1,1/2},\mathbf{C}_{i}\cap
A_{1,1/2})\to 0,$

but either $|a_{i}|>\epsilon$, $d_{H}(\mathbf{C}_{i}\cap
B_{1},\mathbf{C}_{i}^{\prime}\cap B_{1})>\epsilon$, or
$\theta_{\mathbf{C}_{i}}(0)\neq\theta_{\mathbf{C}_{i}^{\prime}}(0)$

Passing to a subsequence, as in Theorem 5.1 we get smooth, multiplicity-one
convergence $\mathbf{C}_{i}\to\mathbf{C}$,
$\mathbf{C}_{i}^{\prime}\to\mathbf{C}^{\prime}$ for
$\mathbf{C},\mathbf{C}^{\prime}\in\mathcal{C}_{\Lambda}$. By Theorem 5.1,
$\theta_{\mathbf{C}_{i}}(0)=\theta_{\mathbf{C}}(0)$ and
$\theta_{\mathbf{C}_{i}^{\prime}}(0)=\theta_{\mathbf{C}^{\prime}}(0)$ for
$i>>1$. In particular, the $\mathbf{C}_{i}$ are planar iff $\mathbf{C}$ is
planar.

Assume first $\mathbf{C}$ is non-planar. If we had
$\limsup_{i}|a_{i}|=\infty$, then after passing to a subsequence Theorem 5.1
implies $a_{i}+\mathbf{C}_{i}^{\prime}$ converges in $C^{2}$ to a plane on
compact subsets of $\mathbb{R}^{8}$, which implies by (28) that
$\mathbf{C}_{i}$ converges to a plane also, which is a contradiction. So we
can assume $a_{i}\to a$ also. From (28) we get
$a+\mathbf{C}^{\prime}=\mathbf{C}$, which implies $a=0$ since
$\theta_{\mathbf{C}}(x)<\theta_{\mathbf{C}}(0)$ for $x\neq 0$. We therefore
have $a=0$, $\mathbf{C}=\mathbf{C}^{\prime}$, which is a contradiction.

Assume $\mathbf{C}$ is a plane. We have

(29) $\displaystyle|a_{i}|$ $\displaystyle=d(0,a+\mathbf{C}^{\prime})$ (30)
$\displaystyle\leq d_{H}((a_{i}+\mathbf{C}_{i}^{\prime})\cap
B_{1},\mathbf{C}\cap B_{1})$ (31) $\displaystyle\leq
cd_{H}((a_{i}+\mathbf{C}_{i}^{\prime})\cap A_{1,1/2},\mathbf{C}\cap
A_{1,1/2})\to 0,$

so $a_{i}\to 0$. As before we deduce $\mathbf{C}^{\prime}=\mathbf{C}$, which
is a contradiction. This proves Lemma 5.3. ∎

The third says that we can regraph over nearby cones.

######  Lemma 5.4.

Given $\Lambda>0$, there is a $\gamma(\Lambda)$ so that if
$\mathbf{C},\mathbf{C}^{\prime}\in\mathcal{C}_{\Lambda}$,
$a\in\mathbb{R}^{8}$, $0\leq\tau,\beta\leq\gamma$ are such that

$|a|\leq\tau,\quad d_{H}(\mathbf{C}^{\prime}\cap\partial
B_{1},\mathbf{C}\cap\partial B_{1})\leq\tau,$

and $v:(a+\mathbf{C}^{\prime})\cap A_{1,1/2}(a)\to\mathbf{C}^{\prime\perp}$ is
a $C^{2}$ function satisfying $|v|_{C^{2}}\leq\beta$ then we can write

$\left(G_{a+\mathbf{C}^{\prime}}(v)\cap A_{1,1/2}(a)\right)\cap
A_{1-a,1/2+a}(0)=G_{\mathbf{C}}(u)\cap A_{1-a,1/2+a}(0),$

for $u:\mathbf{C}\cap A_{1-a,1/2+a}(0)\to\mathbf{C}^{\perp}$ a $C^{2}$
function satisfying $|u|_{C^{2}}\leq c(\Lambda)(\beta+\tau)$.

###### Proof.

Follows by Theorem 5.1:(3),(4) and the inverse function theorem (c.f. Section
8). ∎

##  6\. Cone regions

Here we define various notions of “cone region,” and prove a key smallness
estimate in Theorem 6.3. Ultimately, we will only be interested in strong-cone
regions of hypersurfaces, but we will use cone regions, weak-cone regions, and
their varifold formulations as intermediary constructions. Since we want to
keep track of multiplicity it is most useful to phrase these in terms of
varifolds.

######  Definition 6.0.1 (Weak cone region).

Let $g$ be a $C^{2}$ metric on $B_{R}(a)\subset\mathbb{R}^{8}$, and $V$ an
integral varifold in $(B_{R}(a),g)$. Take $\mathbf{C}\in\mathcal{C}$,
$m\in\mathbb{N}$, $\beta,\tau,\sigma\in[0,1/4]$. We say
$V\llcorner(A_{R,\rho}(a),g)$ is a $(\mathbf{C},m,\beta,\tau,\sigma)$-weak-
cone region111Let us emphasize that, although for ease of notation we write
$A_{R,\rho}(a)$, the actual annulus in which $V$ resembles a cone will be some
perturbation of $A_{R,\rho}(a)$ contained in $A_{R,\rho}(a)$. if the following
occurs: For every $r\in[\rho,(1-\sigma)R)\cap(0,\infty)$, there is an
$a_{r}\in\mathbb{R}^{8}$ with $B_{r}(a_{r})\subset B_{R}(a)$, a cone
$\mathbf{C}_{r}\in\mathcal{C}(\mathbf{C})$, and $C^{2}$ functions
$\\{u_{r,j}:(a_{r}+\mathbf{C}_{r})\cap
A_{r,r/8}(a_{r})\to\mathbf{C}^{\perp}_{r}\\}_{j=1}^{m}$ satisfying

(32) $r^{-1}|u_{r,j}|+|\nabla u_{r,j}|+r|\nabla^{2}u_{r,j}|\leq\beta,\quad
j=1,\ldots,m,$

so that $V$ is a multi-graph:

(33) $V\llcorner
A_{r,r/8}(a_{r})=\sum_{j=1}^{m}[G_{a_{r}+\mathbf{C}_{r}}(u_{r,j})\cap
A_{r,r/8}(a_{r})]_{g},$

and the density ratios are close to constant:

(34) $m\theta_{\mathbf{C}}(0)-\beta\leq\theta_{V}(a_{r},r)\leq
m\theta_{\mathbf{C}}(0)+\beta.$

Additionally, we ask that for all $\rho\leq r\leq
s\leq\min\\{2r,(1-\sigma)R\\}$ we have

(35) $|a_{r}-a_{s}|\leq\tau s,\quad d_{H}(\mathbf{C}_{r}\cap
B_{1},\mathbf{C}_{s}\cap B_{1})\leq\tau.$

When there is ambiguity we may be explicit and write $a_{r}=a_{r}(V)$,
$\mathbf{C}_{r}=\mathbf{C}_{r}(V)$ to specify the varifold in question.

Provided $\beta$, $\tau$, $\sigma$ are sufficiently small, then after
shrinking our radius and enlarging $\beta$, every weak-cone region is actually
a “cone region.”

######  Definition 6.0.2 (Cone region).

In the notation of the above, we say $V\llcorner(A_{R,\rho}(a),g)$ is a
$(\mathbf{C},m,\beta)$-cone region if for every $r\in[\rho,R]\cap(0,\infty)$
there is as $\mathbf{C}_{r}\in\mathcal{C}(\mathbf{C})$ and $C^{2}$ functions
$\\{u_{r,j}:(a+\mathbf{C}_{r})\cap
A_{r,r/8}(a)\to\mathbf{C}_{r}^{\perp}\\}_{j=1}^{m}$ so that (32), (33), (34)
hold with $a_{r}\equiv a$.

A much more non-trivial theorem, and the main theorem of this section, says
that provided $\beta$ is sufficiently small, every cone region is a “strong-
cone” region (for some larger choice of $\beta$).

######  Definition 6.0.3 (Strong cone region).

In the notation of the above, we say $V\llcorner(A_{R,\rho}(a),g)$ is a
$(\mathbf{C},m,\beta)$-strong-cone region if there are $C^{2}$ functions
$\\{u_{j}:(a+\mathbf{C})\cap
A_{R,\rho/8}(a)\to\mathbf{C}^{\perp}\\}_{j=1}^{m}$ so that (32), (33), (34)
hold for every $r\in[\rho,R]\cap(0,\infty)$ with $a_{r}\equiv a$,
$\mathbf{C}_{r}\equiv\mathbf{C}$, and $u_{r,j}\equiv u_{j}$.

It will be convenient to also define cone regions for hypersurfaces:

######  Definition 6.0.4 (Cone regions for $M$).

If $M$ is a $C^{2}$ hypersurface in $B_{R}(a)$, then in the notation of the
above we say $M\cap(A_{R,\rho}(a),g)$ is a
$(\mathbf{C},m,\beta)$-(strong-)cone region if
$[M]_{g}\llcorner(A_{R,\rho}(a),g)$ is a $(\mathbf{C},m,\beta)$-(strong-)cone
region.

A few remarks. If $V\in\mathcal{ISV}_{7}(B_{R}(a),g)$, and
$V\cap(A_{R,\rho}(a),g)$ is a $(\mathbf{C},m,\beta)$-cone region, then $V$ is
regular in $A_{R,\rho}(a)$ by the maximum principle and the fact that
$\dim(\mathrm{sing}V)=0$, and any two graphs in (33) either coincide or are
disjoint. If $\rho=0$, then the maximum principle due to [Ilm96] implies
$V=m[\mathrm{spt}V]_{g}$ in $B_{R}(a)$, and
$[\mathrm{spt}V]_{g}\cap(A_{R,0}(a),g)$ is a $(\mathbf{C},1,\beta)$-cone
region. (33) is just saying that $\mathrm{spt}V\cap A_{r,r/8}(a_{r})$ splits
into at most $m$ conical graphs, but for various reasons it is useful to keep
track of the multiplicity of these graphs.

The following lemma proves that weak-cone regions are effectively cone
regions, after recentering/rescaling, and makes explicit that weak-cone
regions can be recentered, and that extending the recentered region to smaller
inner radii is equivalent to extending the original cone region. These last
facts are important in our contradiction argument later on.

######  Lemma 6.1.

Let $V\llcorner(A_{R,\rho}(a),g)$ be a
$(\mathbf{C},m,\beta,\tau,\sigma)$-weak-cone region, for
$\mathbf{C}\in\mathcal{C}_{\Lambda}$. Assume $\sigma\in[0,1/4]$,
$\tau\leq\sigma/10$. Then:

  1. (1)

We have $|a_{r}-a_{s}|\leq 5\tau\max\\{s,r\\}$ for all
$r,s\in[\rho,(1-\sigma)R]$. Hence, the point

$a_{\rho}=\left\\{\begin{array}[]{l l}a_{\rho}&\rho>0\\\ \lim_{r\to
0}a_{r}&\rho=0\end{array}\right.$

exists, and satisfies $|a_{\rho}-a|\leq 2\sigma R$.

  2. (2)

Take $R^{\prime}\leq(1-2\sigma)R$. Given any
$r\in[\rho,(1-\sigma)R^{\prime}]$, and $\rho^{\prime}\leq r$, then
$V\llcorner(A_{R^{\prime},\rho^{\prime}}(a_{r}),g)$ is a
$(\mathbf{C},m,\beta,\tau,\sigma)$-weak-cone region $\iff$
$V\llcorner(A_{R,\rho^{\prime}}(a),g)$ is a
$(\mathbf{C},m,\beta,\tau,\sigma)$-weak-cone region. In particular,
$V\llcorner(A_{R^{\prime},\rho}(a_{\rho}),g)$ is a
$(\mathbf{C},m,\beta,\tau,\sigma)$-weak-cone region.

  3. (3)

There is a $\gamma(\Lambda)>0$ so that if, additionally,
$\beta,\tau\leq\gamma$, then $V\llcorner A_{(1-3\sigma)R,\rho}(a_{\rho})$ is a
$(\mathbf{C},m,c(\Lambda,m)(\beta+\tau))$-cone region.

###### Proof.

Item 1 follows easily from (35) and (36). We prove Item 2. To prove the
direction $\impliedby$ if will suffice to show that
$B_{R^{\prime}}(a_{r})\subset B_{R}(a)$, and that for every
$s\in[\rho^{\prime},(1-\sigma)R^{\prime}]$ we have $B_{s}(a_{s})\subset
B_{R^{\prime}}(a_{r})$. The first inclusion follows because

(36) $|a-a_{r}|\leq|a-a_{(1-\sigma)R}|+|a_{(1-\sigma)R}-a_{r}|\leq\sigma
R+5\tau(1-\sigma)R\leq 2\sigma R.$

To prove the second inclusion, we observe for any $s$ as above that

$s+|a_{s}-a_{r}|\leq
s+5\tau\max\\{s,r\\}\leq(1+5\tau)(1-\sigma)R^{\prime}<R^{\prime}.$

The direction $\implies$ is trivial if $\rho^{\prime}\geq\rho$. If
$\rho^{\prime}<\rho$, then it suffices to verify that $B_{s}(a_{s})\subset
B_{R}(a)$ for $s\in[\rho^{\prime},\rho]$. But by assumption and (36) we have

$B_{s}(a_{s})\subset B_{R^{\prime}}(a_{r})\subset B_{R}(a).$

This proves Item 2.

To prove Item 3, we first observe that by Item 1, (36) and our assumption
$\tau\leq\sigma/10$, we get for every $r\in[\rho,(1-3\sigma)R]\cap(0,\infty)$:

(37) $\displaystyle B_{r}(a_{\rho})\subset
B_{(1+10\tau)r}(a_{(1+10\tau)r})\subset B_{R}(a),$ (38) $\displaystyle
B_{r}(a_{\rho})\supset B_{(1-5\tau)r}(a_{(1-5\tau)r})\quad\text{ if
}(1-5\tau)r\geq\rho,$ (39)
$\displaystyle\text{and}\quad(1+10\tau)r\leq(1-\sigma)R.$

Therefore, ensuring $\tau(\Lambda),\beta(\Lambda)$ are sufficiently small, for
every such $r$ we can apply Lemma 5.4 in each annulus $A_{r,r/2}(a_{\rho})$ to
get (33), (32).

To get the upper density bound (34), we can use (37), (39) to get for $r$ as
above:

$\displaystyle\theta_{V}(a_{\rho},r)\leq(1+10\tau)^{7}\theta_{V}(a_{(1+10\tau)r},(1+10\tau)r)\leq
m\theta_{\mathbf{C}}(0)+c(\Lambda,m)(\tau+\beta).$

The lower bound is similar, using (38):

$\displaystyle\theta_{V}(a_{\rho},r)$
$\displaystyle\geq(1-5\tau)^{7}\theta_{V}(a_{\max\\{(1-5\tau)r,\rho\\}},\max\\{(1-5\tau)r,\rho\\})$
$\displaystyle\geq m\theta_{\mathbf{C}}(0)-c(\Lambda,m)(\beta+\tau).$

This proves Item 3. ∎

We now work towards proving Theorem 6.3, which is a smallness estimate that
implies cone regions are in fact strong-cone regions. We first require a
helper lemma which implies that small graphicality “propogates” to nearby
scales.

######  Lemma 6.2.

Given $\epsilon,\Lambda>0$, there is a $\gamma(\Lambda)$ so that the following
holds. Let $\mathbf{C}\in\mathcal{C}_{\Lambda}$, $g$ be a $C^{2}$ metric on
$B_{1}$, and $V$ an integral varifold in $(B_{1},g)$. Suppose
$V\llcorner(A_{1,1/2},g)$ is a $(\mathbf{C},m,\beta)$-cone region for
$\beta\leq\gamma$, and

$V\llcorner A_{1,1/2}=\sum_{j=1}^{m}[G_{\mathbf{C}}(u_{j})\cap
A_{1,1/2}]_{g},\quad|u_{j}|_{C^{2}}\leq\epsilon\leq\gamma.$

Then

$V\llcorner A_{1,1/16}=\sum_{j=1}^{m}[G_{\mathbf{C}}(u_{j})\cap
A_{1,1/16}]_{g},\quad|u_{j}|_{C^{2}}\leq c(\Lambda)(\epsilon+\beta).$

###### Proof.

Follows from the definition of cone region and Lemma 5.4. ∎

######  Theorem 6.3.

Given $\epsilon,m,\Lambda>0$, there are $\beta(\Lambda,m,\epsilon)$,
$\delta(\Lambda,m,\epsilon)$ so that the following holds. Let $g$ be a $C^{3}$
metric on $B_{1}$ such that $|g-g_{\mathrm{eucl}}|_{C^{3}(B_{1})}\leq\beta$.
Take $\mathbf{C}\in\mathcal{C}_{\Lambda}$, $m\in\mathbb{N}$, $\rho\in[0,1]$,
and let $V\in\mathcal{ISV}_{7}(B_{1},g)$ be such that
$V\llcorner(A_{1,\rho},g)$ is a $(\mathbf{C},m,\beta)$-cone region. Suppose
there are $C^{2}$ functions $\\{u_{j}:\mathbf{C}\cap
A_{1,1/2}\to\mathbf{C}^{\perp}\\}_{j}$ such that

(40) $V\llcorner A_{1,1/2}=\sum_{j=1}^{m}[G_{\mathbf{C}}(u_{j})\cap
A_{1,1/2}]_{g},\quad|u_{j}|_{C^{2}(\mathbf{C}\cap A_{1,1/2})}\leq\delta.$

Then the $u_{j}$ can be extended to $C^{2}$ functions on $\mathbf{C}\cap
A_{1,\rho/8}$, so that

(41) $V\llcorner A_{1,\rho/8}=\sum_{j=1}^{m}[G_{\mathbf{C}}(u_{j})\cap
A_{1,\rho/8}]_{g},\quad|x|^{-1}|u_{j}|+|\nabla
u_{j}|+|x||\nabla^{2}u_{j}|\leq\epsilon.$

In particular, $V\cap(A_{1,\rho},g)$ is a $(\mathbf{C},m,\epsilon)$-strong-
cone region.

If $\theta_{\mathbf{C}}(0)=1$ then we additionally have the curvature estimate

(42) $\int_{A_{1,\rho/8}}|A_{V,g}|^{7}d\mu_{V}(x)\leq\epsilon$

where $|A_{V,g}|$ is the length of the second fundamental form of
$\mathrm{reg}V\subset(B_{1},g)$.

######  Remark 6.4.

If $V$ is stable in $(B_{1},g)$, then in place of assuming that
$V\llcorner(A_{1,\rho},g)$ is a $(\mathbf{C},m,\beta)$-cone region, one can
instead assume that

(43) $\theta_{V}(0,1)\leq
m\theta_{\mathbf{C}}(0)+\beta,\quad\theta_{V}(0,\rho/16)\geq
m\theta_{\mathbf{C}}(0)-\beta.$

Compare to Theorem 13.1, Lemma 13.2.

######  Remark 6.5.

In place of assuming (34) in Theorem 6.3 one could instead assume directly a
bound of the form

(44)
$\int_{A_{1,\rho/8}}|x|^{-9}|\pi_{V}^{\perp}(x)|^{2}d\mu_{V}(x)\leq\beta.$

In this case it would suffice to know $V$ is only defined and stationary in
the annulus $A_{1,\rho/8}$.

######  Remark 6.6.

By Remark 11.2 (and considering the change of variables $t=\log r$) the same
conclusion in Theorem 6.3 holds if instead of (40) we assume there is a
$\rho^{\prime}\in[\rho/4,1]$ so that

$V\llcorner
A_{\rho^{\prime},\rho^{\prime}/2}=\sum_{j=1}^{m}[G_{\mathbf{C}}(u_{j})\cap
A_{\rho^{\prime},\rho^{\prime}/2}]_{g}$

for some $C^{2}$ functions $u_{j}$ satisfying $\rho^{\prime-1}|u_{j}|+|\nabla
u_{j}|+\rho^{\prime}|\nabla^{2}u_{j}|\leq\delta$.

######  Remark 6.7.

If $g(0)=g_{\mathrm{eucl}}$ and $Dg|_{0}=0$ then we additionally get the Dini-
type estimate

(45) $\int_{A_{1,\rho/8}}|x|^{-8}|\pi_{V}^{\perp}(x)|d\mu_{V}(x)\leq\epsilon,$

which can be viewed as a quantitative version of the statement that tangent
cones are unique. Here (like in (17)) $\pi^{\perp}_{V}|_{x}$ is the Euclidean
orthogonal projection onto the Euclidean normal space $T_{x}^{\perp}V$.

One can recast (45) as a $C^{0}$-Dini-type estimate in the following fashion.
In analogue to the planar Jones’ $\beta$-numbers, let us define the “conical
$\beta$-numbers” $\beta_{\mathbf{C}}(V,a,r)$ by

$\beta_{\mathbf{C}}(V,a,r)=\inf_{\mathbf{C}^{\prime}\in\mathcal{C}(\mathbf{C})}r^{-1}d_{H}(\mathrm{spt}V\cap
A_{r,r/8}(a),(a+\mathbf{C}^{\prime})\cap A_{r,r/8}(a)).$

With $\beta$, $V$ as in Theorem 6.3, there is a $\gamma(\mathbf{C})\geq 2$ so
that if $V\llcorner(A_{1,\rho},g)$ is a $(\mathbf{C},1,\beta)$-strong-cone
region, then $V$ satisfies the estimate

(46)
$\int_{\rho}^{1}\beta_{\mathbf{C}}(V,0,r)^{\gamma}\frac{dr}{r}\leq\epsilon.$

Here $\gamma$ is the $C^{0}$-Lojacsiewicz exponent ([Sim83a, (2.1)]), and if
$\mathbf{C}$ is integrable then one can take $\gamma=2$. In particular, if
$\mathbf{C}$ is planar then

(47) $\int_{A_{1,\rho/8}}|x|^{-5}|A_{V,g}|^{2}d\mu_{V}(x)\leq\epsilon,$

which is a “deeper” reason for estimate (42). If $m\geq 2$, then by the
maximum principle $V\llcorner A_{1,\rho/8}$ splits into $m$ varifolds $V_{j}$,
each of which satisfies (46).

######  Remark 6.8.

By using sharper linear estimates, it probably suffices to assume $g$ is only
$C^{2}$ close to Euclidean (see [Sim84]).

###### Proof.

I claim that it suffices to prove Theorem 6.3 with $\delta$ depending on
$(\mathbf{C},m,\epsilon)$. Let us assume we can do this, and show how to get
$\delta$ to depend only on $(\Lambda,m,\epsilon)$. Let $c_{2}(\Lambda)$,
$\gamma_{2}(\Lambda)$ be the constants from Lemma 5.4, and
$\epsilon_{0}(\Lambda)$ the constant from Theorem 5.1. For each
$\mathbf{C}\in\mathcal{C}_{\Lambda}$, apply (by assumption) Theorem 6.3 with
$\epsilon/(10c_{2})$ in place of $\epsilon$, to obtain a
$\delta_{\mathbf{C}}(\mathbf{C},m,\epsilon)$ and
$\beta_{\mathbf{C}}(\mathbf{C},m,\epsilon)$. WLOG of course we can assume
$\delta_{\mathbf{C}}\leq\min(\epsilon,\gamma_{2},\epsilon_{0})$. Now we can
apply Lemma 5.2 to obtain a finite collection
$\mathbf{C}_{1},\ldots,\mathbf{C}_{\Lambda}$, so that given any
$\mathbf{C}\in\mathcal{C}_{\Lambda}$, we can find an $i$ so that

(48) $d_{H}(\mathbf{C}\cap\partial B_{1},\mathbf{C}_{i}\cap\partial
B_{1})<\delta_{\mathbf{C}_{i}}/(10c_{2})$

Set $\delta=\min_{i=1,\ldots,N}\delta_{\mathbf{C}_{i}}/(10c_{2})$,
$\beta=\min_{i=1,\ldots,N}\beta_{\mathbf{C}_{i}}$.

Now take a general $\mathbf{C}\in\mathcal{C}_{\Lambda}$, and suppose (40)
holds with this $\mathbf{C}$ and $\beta,\delta$ as chosen above. Pick
$\mathbf{C}_{i}$ so that (48) holds, and then by Lemma 5.4 we can write

$V\llcorner A_{1,1/2}=\sum_{j=1}^{m}[G_{\mathbf{C}_{i}}(\tilde{u}_{j})\cap
A_{1,1/2}]_{g},\quad|\tilde{u}_{j}|_{C^{2}}\leq
2c_{2}\delta_{\mathbf{C}_{i}}/(10c_{2})\leq\delta_{\mathbf{C}_{i}}.$

Moreover, by Theorem 5.1 we get that $V\llcorner(A_{1,\rho},g)$ is a
$(\mathbf{C}_{i},m,\beta_{\mathbf{C}})$-cone region. By our choice of
constants we can apply Theorem 6.3 to deduce

$V\llcorner A_{1,\rho/8}=\sum_{j=1}^{m}[G_{\mathbf{C}_{i}}(\tilde{u}_{j})\cap
A_{1,\rho/8}]_{g},\quad|x|^{-1}|\tilde{u}_{j}|+|\nabla\tilde{u}_{j}|+|x||\nabla^{2}\tilde{u}_{j}|\leq\epsilon/(10c_{2}),$

and then finally apply Lemma 5.4 at each scale $r\in[\rho/4,1]$ to get (41).
This proves $\delta$ can be made independent of $\mathbf{C}$.

I next claim that it suffices to prove Theorem 6.3 under the assumption that
$g(0)=g_{\mathrm{eucl}}$, $Dg|_{0}=0$. We prove this. Let $V$, $\mathbf{C}$,
$g$ satisfy the hypotheses of Theorem 6.3. By taking $\phi^{-1}$ to be a
quadratic perturbation of a linear map, we can find a smooth diffeomorphism
$\phi:B_{1/2}\to B_{1}$ satisfying

(49) $|\phi(x)-x|\leq c\beta|x|,\quad|D\phi-Id|+|D^{2}\phi|\leq c\beta,$

so that the pullback metric $g^{\prime}=\phi^{*}g$ satisfies

$g^{\prime}(0)=g_{\mathrm{eucl}},\quad
Dg^{\prime}|_{0}=0,\quad|g^{\prime}-g_{\mathrm{eucl}}|_{C^{3}(B_{1/2})}\leq
c\beta,$

where $c$ is an absolute constant.

From Theorem 5.1, (49), and the inverse function theorem, ensuring
$\beta(\Lambda)$ is sufficiently small, if
$V^{\prime}=(\phi^{-1})_{\sharp}V\llcorner B_{1/2}$ then
$V^{\prime}\in\mathcal{ISV}_{7}(B_{1/2},g^{\prime})$,
$V^{\prime}\llcorner(A_{1/2,2\rho},g^{\prime})$ is a
$(\mathbf{C},m,c(\Lambda)\beta)$-cone region, and (using Lemma 6.2) we can
find $C^{2}$ functions $\\{u_{j}^{\prime}:\mathbf{C}\cap
A_{1/2,1/4}\to\mathbf{C}^{\perp}\\}_{j=1}^{m}$ so that

$V^{\prime}\llcorner
A_{1/2,1/4}=\sum_{j=1}^{m}[G_{\mathbf{C}}(u_{j}^{\prime})\cap
A_{1/2,1/4}]_{g^{\prime}},\quad|u_{j}^{\prime}|_{C^{2}(\mathbf{C}\cap
A_{1/2,1/4})}\leq c(\Lambda)(\delta+\beta).$

For any $\epsilon^{\prime}>0$, ensuring $\beta(\Lambda,m,\epsilon^{\prime})$,
$\delta(\Lambda,m,\epsilon^{\prime})$ are sufficiently small, we can apply
Theorem 6.3 to the rescaled varifold/metric
$(\eta_{0,1/2})_{\sharp}V^{\prime}$, $g^{\prime}\circ\eta_{0,1/2}^{-1}$ to
deduce

$V^{\prime}\llcorner
A_{1/2,\rho/4}=\sum_{j=1}^{m}[G_{\mathbf{C}}(u_{j}^{\prime})\cap
A_{1/2,\rho/4}]_{g^{\prime}},\quad|x|^{-1}|u_{j}^{\prime}|+|\nabla
u_{j}^{\prime}|+|x||\nabla^{2}u_{j}^{\prime}|\leq\epsilon^{\prime}.$

We can again apply the inverse function theorem and Lemma 6.2 to deduce (41)
with $c(\Lambda)(\epsilon^{\prime}+\beta)$ in place of $\epsilon$. Provided
$\epsilon^{\prime}(\Lambda,\epsilon)$, $\beta(\Lambda,\epsilon)$ are
sufficiently small, this proves (41).

If $\theta_{\mathbf{C}}(0)=1$, then since $\phi$ is an isometry we obtain from
the previous paragraph

$\int_{A_{1/4,\rho/2}}|A_{V,g}|^{7}d\mu_{V}\leq\epsilon^{\prime}.$

(42) then follows by (32), provided $\epsilon^{\prime}\leq\epsilon/2$ and
$\beta(m,\epsilon)$ is sufficiently small. This proves my second reduction.

We now prove Theorem 6.3 with $\mathbf{C}$ fixed, and $\delta$,
$\beta\leq\delta$ depending on $(\mathbf{C},m,\epsilon)$, and under the
assumption that $g(0)=g_{\mathrm{eucl}}$, $Dg|_{0}=0$. Note this implies that

(50) $|g|_{x}-g_{\mathrm{eucl}}|\leq\beta|x|^{2},\quad|Dg|_{x}|\leq\beta|x|.$

We note also that there is no loss in assuming $\epsilon(\mathbf{C})$ is as
small as we like. Let $\rho_{*}$ be the least radius for which we have

(51) $V\llcorner A_{1,\rho_{*}}=\sum_{j=1}^{m}[G_{\mathbf{C}}(u_{j})\cap
A_{1,\rho_{*}}]_{g},\quad|x|^{-1}|u_{j}|+|\nabla
u_{j}|+|x||\nabla^{2}u_{j}|\leq\epsilon.$

By Lemma 6.2, ensuring $\delta(\Lambda,\epsilon,m)$,
$\beta(\Lambda,\epsilon,m)$ are sufficiently small, we can assume
$\rho_{*}\leq\max(e^{-6},\rho/8)$. Note also that Lemma 6.2 and our
restriction $\beta\leq\delta$ imply we can assume

(52) $|u_{j}|_{C^{2}(\mathbf{C}\cap A_{1,e^{-2}})}\leq c(\Lambda)\delta.$

We will suppose $\rho_{*}>\rho/8$, and derive a contradiction. By the maximum
principle and our singular set bound we can assume the graphs
$G_{\mathbf{C}}(u_{j})\cap A_{1,\rho_{*}}$ are disjoint, and in particular
each graph $G_{\mathbf{C}}(u_{j})\cap A_{1,\rho_{*}}$ is stationary in
$(A_{1,\rho_{*}},g)$. By this and by Lemma 6.2, it will suffice to show that
for any $0<\epsilon^{\prime}\leq\epsilon$ of our choosing, we can pick any one
$u=u_{j}$ and get the bound

(53) $|x|^{-1}|u|+|\nabla u|+|x||\nabla^{2}u|\leq\epsilon^{\prime}\quad
x\in\mathbf{C}\cap A_{e^{-1},e^{3}\rho_{*}},$

provided $\delta(\mathbf{C},m,\epsilon^{\prime})$,
$\beta(\mathbf{C},m,\epsilon^{\prime})$ are sufficiently small. Our strategy
is to use the decay-growth Theorem 11.1, to show small bounds on $u$ must
persist as long as the density bound (40) holds. We will use notation from
[Sim83a]; see also Appendix 11.

Let $\Sigma=\mathbf{C}\cap\partial B_{1}$, and write $\hat{\nabla}$ for the
connection on $T^{\perp}\Sigma\subset TS^{7}$. Write $r=|x|$, and define the
change of variables $t=-\log r$, $v(t,\theta)=r^{-1}u(r\theta)$, so that $v$
is a $C^{2}$ function on $\Sigma\times(0,T_{*})$ for $T_{*}=-\log\rho_{*}$.
Write $\dot{v}\equiv\partial_{t}v$. Recall the norms $|v|_{k}^{*}(t)$ (defined
in Section 11), and $||v(t)||=||v(t,\cdot)||_{L^{2}(\Sigma)}$. We have

(54) $\frac{1}{c}|v|_{2}^{*}(t)\leq\sup_{x\in e^{-t}\Sigma}|x|^{-1}|u|+|\nabla
u|+|x||\nabla^{2}u|\leq c|v|_{2}^{*}(t)$

for every $t$, and some absolute constant $c$. Define the function
$G:\Sigma\times(\rho_{*},1)\to\mathbb{R}^{8}$ by setting

$G(\theta,r)=\frac{r\theta+u(r\theta)}{\sqrt{1+r^{-1}|u(r\theta)|^{2}}}\equiv\frac{r\theta+rv}{\sqrt{1+|v|^{2}}}$

Provided $\epsilon(\Sigma)$ is sufficiently small, then for any $0\leq a<b\leq
T_{*}$ we can write

$\displaystyle\mathcal{F}_{a,b}(v)$
$\displaystyle:=\mathcal{H}^{7}_{g}(G_{\mathbf{C}}(u)\cap A_{e^{-b},e^{-a}})$
$\displaystyle=\int_{a}^{b}\int_{\Sigma}e^{-7t}(F(\theta,v,\hat{\nabla}v,\dot{v})+H(\theta,t,v,\hat{\nabla}v,\dot{v}))d\theta
dt,$

where the integrand is really just the Jacobian of the map $(\theta,t)\mapsto
G(\theta,e^{-t})$ w.r.t. to the metric $g$, and $e^{-7t}F$ is the Jacobian
w.r.t. $g_{\mathrm{eucl}}$. By embedding $T^{\perp}\Sigma,T\Sigma\subset
T\mathbb{R}^{8}$, one can think of $F(\theta,z,p,q)$, resp.
$H(\theta,t,z,p,q)$, as being defined on
$\Sigma\times\mathbb{R}^{8}\times\mathbb{R}^{64}\times\mathbb{R}^{8}$, resp.
$\Sigma\times\mathbb{R}\times\mathbb{R}^{8}\times\mathbb{R}^{64}\times\mathbb{R}^{8}$
(c.f. [Sim84]).

$F$, $H$ satisfy the following properties: $F(\theta,z,p,q)$ is smooth (and
analytic in $z,p,q$) and independent of $g$, and takes the form

$F(\theta,z,p,q)^{2}=(1+|q|^{2}+q^{2}\cdot z^{2}\cdot
F_{1}(z))E(\theta,z,p)^{2}+q^{2}\cdot p^{2}\cdot F_{2}(\theta,z,p),$

for $F_{1},F_{2}$ smooth, and $E(\theta,z,p)\equiv F(\theta,z,p,0)$; the
functional

$\mathcal{E}(v(t)):=\int_{\Sigma}E(\theta,v,\hat{\nabla}v)\equiv
e^{6t}\mathcal{H}^{6}(G_{\mathbf{C}}(u)\cap\partial B_{e^{-t}})$

satisfies the convexity and analyticity hypotheses of [Sim83a, (1.2), (1.3)],
and has the form

$E(\theta,z,p)^{2}=1+z^{2}\cdot E_{1}(\theta,z,p)+p^{2}\cdot
E_{2}(\theta,z,p)$

for smooth functions $E_{1},E_{2}$; $H(\theta,t,z,p,q)$ is $C^{3}$ in the
$(\theta,t,z)$ variables, smooth in the $(p,q)$ variables, and from (50) $H$
satisfies the bounds

$\sum_{\begin{subarray}{c}0\leq k\leq 3\\\ 0\leq i+j\leq
1\end{subarray}}|D^{k}D_{p}^{i}D_{q}^{j}H|\leq c(\Sigma)\beta e^{-2t},$

here $D$ being the full derivative on
$\Sigma\times\mathbb{R}\times\mathbb{R}^{8}\times\mathbb{R}^{64}\times\mathbb{R}^{8}$.
In the above we are abusing notation slightly: when we write $F_{2}$ (e.g.) we
really mean a collection of functions $F_{2ijkl}^{\alpha\beta}(\theta,z,p)$
defined on $\Sigma\times\mathbb{R}^{8}\times\mathbb{R}^{64}$ so that

$q^{2}\cdot p^{2}\cdot
F_{2}(\theta,z,p)\equiv\sum_{\begin{subarray}{c}i,j,k,l=1,\ldots,8\\\
\alpha,\beta=1,\ldots,8\end{subarray}}q^{i}q^{j}p^{k}_{\alpha}p^{l}_{\beta}F_{2ijkl}^{\alpha\beta}(\theta,z,p)$

A straightforward computation then shows that stationarity of $v$ for the area
functional $\mathcal{F}_{a,b}$ implies that $v$ solves a PDE of the form

(55)
$\displaystyle-\ddot{v}+7\dot{v}-\mathcal{M}(v)-\mathcal{R}_{1}(v)-\mathcal{R}_{2}(v)=f$

where, as in Section 11, $\mathcal{M}=-\mathrm{grad}\mathcal{E}$ in the
$L^{2}(\Sigma)$ sense; $\mathcal{R}_{1}$ is independent of $g$ and has the
form of Section 11:Item 2; and $\mathcal{R}_{2}$, $f$ are as in Section
11:Items 1, 3, except with $c(\Sigma)\beta e^{-2t}$ in place of $\delta
e^{-\epsilon t}$.

As exploited in [Sim83a], for any $\delta^{\prime}>0$ provided
$\beta(\Sigma,\delta^{\prime})$ and $\epsilon(\Sigma,\delta^{\prime})$ are
sufficiently small then both $v$ and $w=\dot{v}$ solves an PDE of the form

(56)
$\displaystyle-\ddot{w}+7\dot{w}-Lw+e_{1}\cdot\hat{\nabla}^{2}w+e_{2}\cdot\hat{\nabla}\dot{w}+e_{3}\ddot{w}+e_{4}\cdot\hat{\nabla}w+e_{5}\dot{w}+e_{6}w=f^{\prime}$

for $e_{i}(x,t)$, $f^{\prime}$ being $C^{1}$ functions (different for $v$ and
$w$) having the estimates $|e_{i}|_{1}^{*}(t)\leq\delta^{\prime}$,
$|f^{\prime}|_{1}^{*}(t)\leq c(\Sigma)\beta e^{-2t}$, and $L$ being the
(elliptic) linearization of $\mathcal{M}$ at $0$. In fact, for us
$L=\hat{\Delta}_{\Sigma}+|A_{\Sigma}|^{2}+6$ where $A_{\Sigma}$ is the second
fundamental form of $\Sigma\subset S^{7}$, and $\hat{\Delta}_{\Sigma}$ the
connection Laplacian on $T^{\perp}\Sigma$. Ensuring $\delta^{\prime}(\Sigma)$
is small, we can apply standard elliptic theory to get

(57) $\displaystyle|v|_{2}(t)^{*}\leq
c(\Sigma)(||v||_{L^{1}(\Sigma\times(t-1,t+1))}+\beta e^{-2t})\quad
t\in[1,T_{*}-2],$

and the same for $\dot{v}$.

The key that allows us to apply Theorem 11.1 to $v$ is the “Hardt-Simon” type
inequality (c.f. [Sim93]): provided $\epsilon(\Sigma)$ is sufficiently small,
then we have

(58) $|\dot{v}(t,\theta)|^{2}\leq
2e^{2t}|\pi^{\perp}_{V}(G(\theta,e^{-t}))|^{2}\leq
4|\dot{v}(t,\theta)|^{2},\quad\forall t\in(0,T_{*}),\theta\in\Sigma,$

where $\pi^{\perp}_{V}$ is the Euclidean projection onto the Euclidean normal
direction $T^{\perp}_{G(\theta,r)}V$. This follows by first observing that
$r\mapsto G(r,\theta)$ is a curve in $\mathrm{reg}V$, and hence

$-\frac{\pi^{\perp}_{V}(r\partial_{r}v)}{\sqrt{1+|v|^{2}}}=(1-(r\partial_{r}v)\cdot(G/r))\pi^{\perp}_{V}(G/r),$

and then second observing that we have the bound
$|\pi^{\perp}V-\pi^{\perp}_{T_{r\theta}\mathbf{C}}|\leq c(\Sigma)\epsilon$.

Using the monotonicity (17), (50), our hypothesis on $g$, and observing that
the Jacobian of $(r,\theta)\mapsto G(\theta,r)$ is
$\geq(1-c(\Sigma)(\epsilon+\beta))r^{6}$, we can use (58) to estimate:

$\displaystyle\int_{t}^{s}||\dot{v}(t)||^{2}dt$ $\displaystyle\leq
2\int_{e^{-s}}^{e^{-t}}\int_{\Sigma}r^{-3}|\pi^{\perp}_{V}(G(r,\theta))|^{2}d\theta
dr$ $\displaystyle\leq 4\int_{A_{e^{-s},e^{-t}}\cap
V}\frac{|\pi^{\perp}_{V}(x)|^{2}}{|x|^{7+2}}d\mu_{V}(x)$ $\displaystyle\leq
c(\Lambda)\beta,$

provided $\epsilon(\Sigma),\beta(\Sigma)$ are sufficiently small, and
$0<s<t\leq T_{*}-3$

We deduce that, for $s,t$ as above, we have

(59) $||v(t)-v(s)||\leq\int_{s}^{t}||\dot{v}(t)||dt\leq
c(\Lambda)\beta^{1/2}|t-s|^{1/2}.$

Therefore, combining (59) with estimates (57), we obtain for any
$s,t\in[1,T_{*}-3]$ the bounds

$\displaystyle|v|_{2}^{*}(t)$ $\displaystyle\leq
c(\Sigma)(\sup_{t^{\prime}\in(t-1,t+1)}||v(t)||+\beta)$ (60)
$\displaystyle\leq
c(\Sigma)|v|_{2}^{*}(s)+c(\Sigma)\beta^{1/2}\max\\{1,|t-s|\\},$

and hence $v$ has $(c(\Sigma),c(\Sigma)\beta^{1/2})$-linear growth on
$\Sigma\times[1,T_{*}-3]$.

On the other hand, we can use (55), the $C^{1}$ Lojasiewicz-Simon inequality
for $\mathcal{E}$ [Sim83a, (2.2)], and (57) to get

$\displaystyle|\mathcal{E}(v(t))-\mathcal{E}(0)|^{1-\theta}$
$\displaystyle\leq||\mathcal{M}(v(t))||$ $\displaystyle\leq
c(\Sigma)|\dot{v}|_{1}^{*}(t)+c(\Sigma)\beta e^{-2t}$ $\displaystyle\leq
c(\Sigma)||\dot{v}||_{L^{2}(\Sigma\times(t-1,t+1))}+c(\Sigma)\beta e^{-2t}$
(61) $\displaystyle\leq c(\Sigma)\beta^{1/2},$

for any $t\in[1,T_{*}-3]$. Here $\theta(\Sigma)\in(0,1/2)$ is the Lojasiewicz
exponent as in [Sim83a, Theorem 3].

Our hypothesis (52), and (60), (61), imply that provided $\delta(\Sigma)$,
$\beta(\delta,\Sigma)$ are sufficiently small, we can apply Theorem 11.1 to
deduce that

(62) $|v|_{2}^{*}(t)\leq c(\Sigma)\delta^{\alpha}\quad\forall
t\in[1,T_{*}-3],$

for some $\alpha(\Sigma)\in(0,1/2)$. In particular, by (54), ensuring
$\delta(\Sigma,\epsilon^{\prime})$ is small, we deduce (53). From Theorem 11.1
we also have the bound

(63) $\int_{1}^{T_{*}-3}||\dot{v}(t)||dt\leq c(\Sigma)\delta^{\alpha/2}$

which together with (62), (58) implies (45).

Using a similar computation to (61), the $C^{0}$ Lojasiewicz-Simon inequality
[Sim83a, (2.1)] implies there is a $\gamma(\Sigma)\geq 2$, so that for every
$t\in[1,T_{*}-3]$ we can find a $\zeta_{t}\in C^{2}(\Sigma,\Sigma^{\perp})$
solving $\mathcal{M}(\zeta_{t})=0$ so that

$\displaystyle||v(t)-\zeta_{t}||^{\gamma}$ $\displaystyle\leq
2||\mathcal{M}(v(t))||$ $\displaystyle\leq
c(\Sigma)||\dot{v}||_{L^{1}(\Sigma\times(t-1,t+1))}+c(\Sigma)\beta e^{-2t}$
$\displaystyle\leq c(\Sigma)\int_{t-1}^{t+1}||\dot{v}(t)||dt+c(\Sigma)\beta
e^{-2t}.$

Here $\gamma$ is the Lojasiewicz exponent, and we note that if $\mathbf{C}$ is
“‘integrable” then one can take $\gamma=2$ (see e.g. [Sim96, Page 80]).

If $2\leq i\leq T_{*}-7$, then by (59), the above, and (62) (since $\gamma\geq
2$), we get

$\sup_{t\in[i,i+3]}||v(t)-\zeta_{i}||^{\gamma}\leq
c(\Sigma)\int_{i-1}^{i+4}||\dot{v}(t)||dt+c(\Sigma)\beta e^{-2t}.$

Setting $\zeta_{i}=0$ if $i\leq 2$ or $i>T_{*}-7$, we deduce there are numbers
$\\{\epsilon_{i}\\}_{i}$ so that every integer $i\in(0,T_{*})$ and for every
$t\in[0,T_{*})\cap[i-3,i]$, we have

(64)
$\displaystyle||v(t)-\zeta_{i}||^{\gamma}\leq\epsilon_{i},\quad\sum_{i}\epsilon_{i}\leq
c(\Sigma)\delta^{\alpha/2}.$

When $\mathbf{C}$ is planar, this in particular implies there are planes
$\mathbf{C}_{i}$ so that

$\left(e^{i}d_{H}(G_{\mathbf{C}}(u)\cap A_{e^{-i},e^{-i-3}},\mathbf{C}_{i}\cap
A_{e^{-i},e^{-i-3}})\right)^{2}\leq\epsilon_{i},$

for $i,\epsilon_{i}$ as in (64). Hence, ensuring $\beta$ is sufficiently
small, by (14) we deduce that for every $r\in[2\rho,1/2]$ we have the bound

$\displaystyle\sup_{G_{\mathbf{C}}(u)\cap A_{r,r/8}}r^{2}|A|^{2}$
$\displaystyle\leq
c|g-g_{\mathrm{eucl}}|^{2}_{C^{0}(B_{2r})}+cr^{2}|Dg|^{2}_{C^{1}(B_{2r})}+c\epsilon_{i}$
$\displaystyle\leq c\beta r^{2}+c\epsilon_{i},$

for $c$ an absolute constant, and $A$ be the second fundamental form of
$G_{\mathbf{C}}(u)\subset(B_{1},g)$. Together with (64), (19), this implies

$\int_{A_{1/2,\rho/4}}|x|^{-5}|A|^{2}d\mu_{V}\leq
c(m)(\beta+\delta^{\alpha/2}),$

which combined with (32) and Theorem 5.1 gives (47), (42) provided
$\delta(m,\epsilon),\beta(m,\epsilon)$ are chosen sufficiently small. ∎

##  7\. Decomposition

Here is the core of our argument. We show that a minimal surface lying
sufficiently close to a stable minimal cone with multiplicity must break up
into a “cone decomposition,” consisting of a controlled number of “smooth
regions” (multi-graphs over a smooth model surface) and a controlled number of
“strong cone regions” (multi-graphs over cones). We already defined cone
regions in the previous section. We first define our smooth models/regions and
cone decompositions. In this section we work in $\mathbb{R}^{8}$.

A smooth model is an entire minimal hypersurface-with-multiplicity $S$ in
$\mathbb{R}^{8}$ along with a choice of disjoint balls, so that $S$ is
entirely smooth outside of these balls. The multiplicities of $S$ specify the
number of graphs we expect to see over each component. Each hole more or less
corresponds to a would-be singularity or point of index concentration.

######  Definition 7.0.1 (Smooth model).

Given $\Lambda,\gamma\geq 0$, $\sigma\in(0,1/3)$, we say that
$(S,\mathbf{C},\\{(\mathbf{C}_{\alpha},B_{r_{\alpha}}(y_{\alpha}))\\}_{\alpha})$
is a $(\Lambda,\sigma,\gamma)$-smooth model if
$S\in\mathcal{ISV}_{7}(\mathbb{R}^{8},g_{\mathrm{eucl}})$ and satisifes
$\theta_{S}(0,\infty)\leq\Lambda$, $\mathbf{C}\in\mathcal{C}_{\Lambda}$,
$\\{B_{2r_{\alpha}}(y_{\alpha})\\}_{\alpha}$ is a finite collection of
disjoint balls in $B_{1-3\sigma}$, and $\\{\mathbf{C}_{\alpha}\\}_{\alpha}$ is
a corresponding finite collection of cones in $\mathcal{C}_{\Lambda}$, such
that the following is satisfied:

  1. (1)

There is a decomposition

(65) $S=m_{1}[S_{1}]+\ldots+m_{k}[S_{k}],$

where $m_{i}\in\mathbb{N}$, and $S_{i}$ are smooth, disjoint, closed (as
sets), embedded minimal hypersurfaces in
$\mathbb{R}^{8}\setminus\cup_{\alpha}y_{\alpha}$.

  2. (2)

There is an $m\in\mathbb{N}$ satisfying $m\theta_{\mathbf{C}}(0)\leq\Lambda$
so that $S\llcorner(A_{\infty,1},g_{\mathrm{eucl}})$ is a
$(\mathbf{C},m,\gamma)$-strong-cone region.

  3. (3)

For each $\alpha$, there is a $j$ so that $\mathrm{spt}S\cap
A_{2r_{\alpha},0}(y_{\alpha})=S_{j}\cap A_{2r_{\alpha},0}(y_{\alpha})$ and
$S_{j}\cap(A_{2r_{\alpha},0}(y_{\alpha}),g_{\mathrm{eucl}})$ is a
$(\mathbf{C}_{\alpha},1,\gamma)$-strong-cone region.

Often we will refer to the smooth model as simply $S$, where the existence of
the associated
$\mathbf{C},\\{(\mathbf{C}_{\alpha},B_{r_{\alpha}}(y_{\alpha}))\\}_{\alpha}$
is implicitly understood.

Up to scaling, any $S\in\mathcal{ISV}_{7}(\mathbb{R}^{8},g_{\mathrm{eucl}})$
with $\theta_{S}(0,\infty)\leq\Lambda$ and
$\mathrm{index}(S,\mathbb{R}^{8},g_{\mathrm{eucl}})<\infty$ is a smooth model
for some choice of balls $\\{B_{r_{\alpha}}(y_{\alpha})\\}_{\alpha}$. In some
sense the main content of this definition is the choice of balls, as pinning
down a scale by which to measure $S$, which is made precise in the following
definition.

######  Definition 7.0.2.

Given a smooth $(\Lambda,\sigma,\gamma)$-model $S$, we let $\epsilon_{S}$ be
the largest number $\leq\min\\{1,\min_{\alpha}\\{r_{\alpha}\\}\\}$ for which
the map $G_{S}:T^{\perp}(\cup_{j}S_{j})\to\mathbb{R}^{8}$ defined by
$G_{S}(x,v)=x+v$ is a diffeomorphism from $\\{(x,v)\in
T^{\perp}(\cup_{j}S_{j}):x\in
B_{2}\setminus\cup_{\alpha}B_{r_{\alpha}/8}(y_{\alpha}),|v|<2\epsilon_{S}\\}$
onto its image, and satisfies $|DG_{S}|_{(x,t)}-Id|\leq\epsilon_{S}^{-1}|t|$.
Note that $\epsilon_{S}$ is always positive.

A smooth region is a multi-graph over $S$, with $C^{2}$ norm small at the
scale of $\epsilon_{S}$.

######  Definition 7.0.3 (Smooth region).

Given a smooth model $S$, $g$ a $C^{2}$ metric on
$B_{R}(a)\subset\mathbb{R}^{8}$, and $V$ an integral varifold in
$(B_{R}(a),g)$, we say $V\llcorner(B_{R}(a),g)$ is a $(S,\beta)$-smooth region
if for each $i=1,\ldots,k$ there are $C^{2}$ functions $\\{u_{ij}:S_{i}\to
S_{i}^{\perp}\\}_{j=1}^{m_{i}}$ so that

$\left((\eta_{a,R})_{\sharp}V\right)\llcorner
B_{1}\setminus\cup_{\alpha}B_{r_{\alpha}/4}(y_{\alpha})=\sum_{i=1}^{k}\sum_{j=1}^{m_{i}}[\mathrm{graph}_{S_{i}}(u_{ij})\cap
B_{1}\setminus\cup_{\alpha}B_{r_{\alpha}/4}(y_{\alpha})]_{g\circ\eta_{a,R}^{-1}},$

and

$|u_{ij}|_{C^{2}(S_{i})}\leq\beta\epsilon_{S},\quad\forall i,j.$

The “global” structure of one of our minimal surfaces/stationary varifolds is
captured by a strong-cone decomposition, which is essentially a finite
collection of strong-cone regions and smooth regions which fit together to
cover all of $B_{R}(x)$. The total number of regions is controlled, and the
smooth models can only come from some (finite) predefined pool of models
$\mathcal{S}$. Similar structures have appeared in many other contexts,
sometimes called bubble trees (e.g. for harmonic maps [Par96]) or neck
decompositions (e.g. for limits of spaces with bounded Ricci [JN21]).

######  Definition 7.0.4 (Cone decomposition).

Take $\theta,\gamma,\beta\in\mathbb{R}$, $\sigma\in(0,1/3)$, $N\in\mathbb{N}$.
Let $g$ be a $C^{2}$ metric on $B_{R}(x)\subset\mathbb{R}^{8}$, $V$ be an
integral varifold in $(B_{R}(x),g)$, and let $\mathcal{S}=\\{S_{s}\\}_{s}$ be
a finite collection of $(\theta,\sigma,\gamma)$-smooth models. A
$(\theta,\beta,\mathcal{S},N)$-cone decomposition of $V\llcorner(B_{R}(x),g)$
consists of integers $N_{S},N_{C}$, satisfying $N_{C}+N_{S}\leq N$, points
$\\{x_{a}\\}_{a},\\{x_{b}\\}_{b}\subset B_{R}(x)$, radii $\\{R_{a}\geq
2\rho_{a}\\}_{a}$, $\\{R_{b}\\}_{b}$, indices $\\{s_{b}\\}_{b}$, integers
$\\{1\leq m_{a}\leq[\theta]\\}_{a}$, and cones
$\\{\mathbf{C}_{a}\\}_{a}\subset\mathcal{C}$, where $a=1,\ldots,N_{C}$,
$b=1,\ldots,N_{S}$, such that:

  1. (1)

Every $V\llcorner(B_{R_{b}}(x_{b}),g)$ is a $(S_{s_{b}},\beta)$-smooth region,
and every $V\llcorner(A_{R_{a},\rho_{a}}(x_{a}),g)$ is a
$(\mathbf{C}_{a},m_{a},\beta)$-strong-cone region.

  2. (2)

There is either a smooth region $B_{R_{b}}(x_{b})$ with $R_{b}=R$, $x_{b}=x$,
or a strong-cone region $A_{R_{a},\rho_{a}}(x_{a})$ with $R_{a}=R$, $x_{a}=x$.

  3. (3)

If $B_{R_{b}}(x_{b})$ is a smooth region, with associated smooth model
$(S,\mathbf{C},\\{(\mathbf{C}_{\alpha},B_{r_{\alpha}}(y_{\alpha}))\\}_{\alpha})\in\mathcal{S}$,
then for each $\alpha$ there we can find a radius $R_{b,\alpha}$ and point
$x_{b,\alpha}$ satisfying

$|x_{b,\alpha}-y_{\alpha}|\leq\beta
R_{b}r_{\alpha},\quad\frac{1}{2}\leq\frac{R_{b,\alpha}}{R_{b}r_{\alpha}}\leq
1+\beta,$

so that there is either a strong-cone region $A_{R_{a},\rho_{a}}(x_{a})$ with
$R_{a}=R_{b,\alpha}$, $x_{a}=x_{b,\alpha}$, or another smooth region
$B_{R_{b^{\prime}}}(x_{b^{\prime}})$ with $R_{b^{\prime}}=R_{b,\alpha}$,
$x_{b^{\prime}}=x_{b,\alpha}$.

  4. (4)

If $A_{R_{a},\rho_{a}}(x_{a})$ is a $(\mathbf{C}_{a},m_{a},\beta)$-strong-cone
region, and $\rho_{a}>0$, then there is either a smooth region
$B_{R_{b}}(x_{b})$ with $R_{b}=\rho_{a}$, $x_{b}=x_{a}$, or another cone
region $A_{R_{a^{\prime}},\rho_{a^{\prime}}}(x_{a^{\prime}})$ with
$R_{a^{\prime}}=\rho_{a}$, $x_{a^{\prime}}=x_{a}$. If $\rho_{a}=0$, then
$\theta_{\mathbf{C}_{a}}(0)>1$.

Later on we will be primarily interested in cone decompositions for
hypersurfaces. We define them in terms of the associated (multiplicity-one)
varifold.

######  Definition 7.0.5 (Smooth region/Cone decomposition for $M$).

Given a $C^{2}$ hypersurface $M\subset B_{R}(a)$, and $g$ a $C^{2}$ metric on
$B_{R}(a)$, we say $M\cap(B_{R}(x),g)$ is a $(S,\beta)$-smooth region or
admits a $(\theta,\beta,\mathcal{S},N)$-cone decomposition if
$[M]_{g}\llcorner(B_{R}(a),g)$ is a $(S,\beta)$-smooth region or admits a
$(\theta,\beta,\mathcal{S},N)$-cone decomposition (respectively).

Recall that Theorem 5.1 gives an enumeration of
$\\{\theta_{\mathbf{C}}(0):\mathbf{C}\in\mathcal{C}\\}=\\{\theta_{i}\\}_{i}$.
Let us enumerate the set of densities-with-multiplicity as
$\\{m\theta_{i}:m\in\mathbb{N},i=0,1,\ldots\\}=\\{\tilde{\theta}_{i}\\}_{i}$,
for $1=\tilde{\theta}_{0}<\tilde{\theta}_{1}<\ldots$. We prove our
decomposition theorem $(\dagger_{l,I})$ by induction on both the density-with-
multplicity $\tilde{\theta}_{l}$ and index $I$.

######  Theorem 7.1 ($\dagger_{l,I}$).

Given $l,I\in\mathbb{N}$, $0<\beta\leq\gamma\leq 1$,
$\sigma\in(0,\frac{1}{100(I+1)}]$, there are constants $\delta_{l,I}$, $N$,
and a finite collection of $(\tilde{\theta}_{l},\sigma,\beta)$-smooth models
$\\{S_{s}\\}_{s}=\mathcal{S}$, all depending (only) on
$(l,I,\gamma,\beta,\sigma)$, so that the following holds.

Let $g$ be a $C^{3}$ metric on $B_{1}$ satisfying
$|g-g_{\mathrm{eucl}}|_{C^{3}(B_{1})}\leq\delta_{l,I}$. Take
$V\in\mathcal{ISV}_{7}(B_{1},g)$ with $\mathrm{index}(V,B_{1},g)\leq I$,
$\mathbf{C}\in\mathcal{C}$, $m\in\mathbb{N}$, such that
$m\theta_{\mathbf{C}}(0)\leq\tilde{\theta}_{l}$. Suppose that

(66) $d_{H}(\mathrm{spt}V\cap B_{1},\mathbf{C}\cap B_{1})\leq\delta_{l,I},$

and

(67)
$(m-1/2)\theta_{\mathbf{C}}(0)\leq\theta_{V}(0,1/2),\quad\theta_{V}(0,1)\leq(m+1/2)\theta_{\mathbf{C}}(0).$

Then there is a radius $r\in(1-20(I+1)\sigma,1)$, so that
$V\llcorner(B_{r},g)$ admits a
$(\tilde{\theta}_{l},\beta,\mathcal{S},N)$-strong-cone decomposition. If $I=0$
then one can take $r=1-5\sigma$.

Additionally, each $S_{s}$ satisfies $\mathrm{index}(S_{s})\leq I$. If $V$ is
area-minimizing,222In the sense that $V$ is the varifold associated to some
mass-minimizing current. This statement is a direct consequence of the nature
of the proof, and the compactness for mass-minimizing currents [Sim83b,
Chapter 7, Theorem 2.4]. then one can assume all the $S_{s}$ appearing in its
cone decomposition are area-minimizing also.

######  Remark 7.2.

We emphasize that $\delta_{l,I}$ depends also on $\gamma,\beta,\sigma$. We
write only the indices $l,I$ as these are the integers we induct over.

######  Remark 7.3.

By the curvature estimate (42), if all the cones appearing in the cone
decomposition of $V$ are planar, then $\mathrm{sing}V=\emptyset$ and we have
the bound

$\int_{B_{r}}|A_{V,g}|^{7}d\mu_{V}(x)\leq c(l,\mathcal{S},N).$

######  Remark 7.4.

We will often use Theorem 7.1 at some different scale $B_{r}(x)$. Let us
mention explicitly that if $g$ is a $C^{3}$ metric on
$B_{r}(x)\subset\mathbb{R}^{8}$ such that
$|g-g_{\mathrm{eucl}}|_{C^{3}(B_{r}(x))}\leq\delta$, then
$g\circ\eta_{x,r}^{-1}\equiv g(x+r\cdot)$ is a $C^{3}$ metric on $B_{1}$
satisfying
$|g\circ\eta_{x,r}^{-1}-g_{\mathrm{eucl}}|_{C^{3}(B_{1})}\leq\max\\{1,r\\}\delta$.

######  Remark 7.5.

The cone decomposition constructed by Theorem 7.1 has the additional
properties: the outermost region is always a smooth region, and no two cone
regions are “stacked” directly inside each other (so you cannot have large
annulus consisting of numerous cone regions where $V$ looks like a rotating
cone).

###### Proof.

Fix $l,I$. We shall prove $(\dagger_{l,I})$ by induction and by contradiction.
We shall choose our constants as we go along, but it may help the reader to
keep in mind:

$\beta^{\prime\prime}<<\tau<<\beta^{\prime}<<\beta\leq\gamma<<\sigma<1.$

Note there is no loss in assuming $\beta(l,\sigma)$ is as small as we like.

Suppose $(\dagger_{l,I})$ fails. Then there are sequences $\delta_{i}\to 0$,
$C^{3}$ metrics $g_{i}$, $V_{i}\in\mathcal{ISV}_{7}(B_{1},g_{i})$,
$\mathbf{C}_{i}\in\mathcal{C}$, $m_{i}\in\mathbb{N}$, so that $g_{i}$, $V_{i}$
satisfies the hypotheses of $(\dagger_{l,I})$ with $\delta_{i}$, $g_{i}$,
$m_{i}$, $\mathbf{C}_{i}$ in place of $\delta,g,m,\mathbf{C}$, but with the
property that for any finite collection $\mathcal{S}^{\prime}$ of
$(\tilde{\theta}_{l},\sigma,\beta)$-smooth models, and any
$N^{\prime}\in\mathbb{N}$, there is an $i_{0}$ such that
$V\llcorner(B_{r},g_{i})$ does not admit a
$(\tilde{\theta}_{l},\beta,\mathcal{S}^{\prime},N^{\prime})$-strong-cone
decomposition for all $i>i_{0}$, $r\in(1-20(I+1)\sigma,1)$.

Passing to a subsequence, by Theorem 5.1 we can assume that
$\mathbf{C}_{i}\to\mathbf{C}\in\mathcal{C}$ smoothly with multiplicity-one
away from $0$, and $\theta_{\mathbf{C}_{i}}(0)=\theta_{\mathbf{C}}(0)$ for all
$i$. By Lemmas 4.4, 4.2, and our hypotheses, passing to a further subsequence,
there are at most $I$ points $\mathcal{I}\subset\mathbf{C}\cap B_{1}$ so that
$V_{i}\to m[\mathbf{C}]$ as varifolds in $B_{1}$, in $C^{2}$ on compact
subsets of $\mathcal{I}\cup\mathrm{sing}\mathbf{C}$. Note that since
$m\theta_{\mathbf{C}}(0)\leq\tilde{\theta}_{l}$, we have
$\max\\{m,\theta_{\mathbf{C}}(0)\\}\leq\tilde{\theta}_{l}$.

We first note that if $l=0$, corresponding to $m[\mathbf{C}]=[\mathbf{C}]$
being a multiplicity-one plane, and $I$ is arbitrary, then provided $i>>1$
[All72] proves implies each $V_{i}\llcorner(B_{1-5\sigma},g_{i})$ is a
$(S,\beta)$-smooth region, for $S$ being the smooth model
$([\mathbf{C}],\mathbf{C},\emptyset)$. Similarly, if $\mathbf{C}$ is planar,
$I=0$, and $l=m$ is arbitrary, then since $\mathrm{reg}V$ is stable in
$(B_{1},g_{i})$ [SS81] implies $V_{i}\llcorner(B_{1-5\sigma},g_{i})$ is a
$(S,\beta)$-smooth region for $i>>1$, where $S$ is the smooth model
$(m[\mathbf{C}],\mathbf{C},\emptyset)$. In either case we get that
$V_{i}\llcorner(B_{1-5\sigma},g_{i})$ admits a
$(\tilde{\theta}_{l},\beta,\\{S\\},1)$-cone decomposition, which is a
contradiction.

By our inductive hypothesis we can therefore assume that
$(\dagger_{l^{\prime},I})$ holds for every $l^{\prime}<l$, $I$ arbitrary, and
also that $(\dagger_{l,I^{\prime}})$ holds for any $I^{\prime}<I$.

We break into three cases.

Case 1: $\mathcal{I}\subset\\{0\\}$. For each $i$, let $\rho_{i}$ be the
smallest number such that $V_{i}\llcorner(A_{1,\rho_{i}}(0),g_{i})$ is a
$(\mathbf{C},m,\beta^{\prime\prime},\tau,\sigma)$-weak-cone region, for
$\beta^{\prime\prime},\tau$ to be chosen later depending only on
$(l,\beta,\gamma)$. Let $a_{i}=a_{\rho_{i}}(V_{i})$ be the annulus center at
radius $\rho_{i}$ (as in Definition 6.0.1). By our varifold convergence
$V_{i}\to m[\mathbf{C}]$, and $C^{2}$ convergence on compact subsets of
$B_{1}\setminus\\{0\\}$, we have $a_{i}\to 0$ and $\rho_{i}\to 0$.

Ensuring $\beta^{\prime\prime}(l,\beta^{\prime})$,
$\tau(l,\beta^{\prime},\sigma)$ are sufficiently small, Lemma 6.1 implies each
$V_{i}\llcorner(A_{1-3\sigma,\rho_{i}}(a_{i}),g_{i})$ is a
$(\mathbf{C},m,\beta^{\prime})$-cone region. Ensuring
$\beta^{\prime}(l,\beta)$ is sufficiently small, and $i>>1$, we can apply
Theorem 6.3 to deduce that each
$V_{i}\llcorner(A_{1-3\sigma,\rho_{i}}(a_{i}),g_{i})$ is a
$(\mathbf{C},m,\beta)$-strong-cone region.

First suppose $\rho_{i}=0$ for infinitely-many $i$. For each such $i>>1$, the
maximum principle of [Ilm96] implies $V_{i}\llcorner
B_{1-4\sigma}=m[M_{i}]_{g_{i}}$ for some
$M_{i}\in\mathcal{M}_{7}(B_{1-4\sigma},g_{i})$. If $\theta_{\mathbf{C}}(0)=1$,
corresponding to $m[\mathbf{C}]$ being a multiplicity-$m$ plane, then
$[M_{i}]_{g_{i}}\to[\mathbf{C}]$ in $B_{1-4\sigma}$ and so by [All72] we
deduce $V_{i}\llcorner(B_{1-5\sigma},g_{i})$ is a $(S,\beta)$-smooth region
for $S$ the smooth model $(m[\mathbf{C}],\mathbf{C},\emptyset)$. So
infinitely-many $V_{i}\llcorner(B_{1-5\sigma},g_{i})$ admit a
$(\tilde{\theta}_{l},\beta,\\{S\\},1)$-cone decomposition, which is a
contradiction.

If $\theta_{\mathbf{C}}(0)>1$, define the smooth model
$(S=[\mathbf{C}],\mathbf{C},\\{(\mathbf{C},B_{1/10})\\})$, and let
$r=1-5\sigma$. By $C^{2}$ convergence, $V_{i}\llcorner(B_{r},g_{i})$ is a
$(S,\beta)$-smooth region for $i>>1$. Since $|a_{i}|\leq\beta r/10$ and
$V_{i}\llcorner(A_{r/10,\rho_{i}}(a_{i}),g_{i})$ is a
$(\mathbf{C},m,\beta)$-strong-cone region for every $i>>1$, we deduce that
infinitely-many $V_{i}\llcorner(B_{r},g_{i})$ admit a
$(\tilde{\theta}_{l},\beta,\\{S\\},2)$-strong-cone decomposition. This is a
contradiction.

So we must have $\rho_{i}>0$ for all $i>>1$. Define the rescaled varifolds
$V_{i}^{\prime}=(\eta_{a_{i},\rho_{i}})_{\sharp}V_{i}$, and rescaled metrics
$g_{i}^{\prime}=g_{i}\circ\eta_{a_{i},\rho_{i}}^{-1}$. Then for any $R>2$ and
$i>>1$, $V_{i}^{\prime}\in\mathcal{ISV}_{7}(B_{R}(0),g_{i}^{\prime})$, and by
Lemma 6.1(2) and scale-invariance (Remark 4.3), we have that

(68)
$1=\inf\\{\rho:V_{i}^{\prime}\llcorner(A_{R,\rho}(0),g_{i}^{\prime})\text{ is
a $(\mathbf{C},m,\beta^{\prime\prime},\tau,\sigma)$-weak-cone region}\\}.$

Of course we also have that $a_{1}(V_{i}^{\prime})=0$, this being the center
of the annular region of $V_{i}^{\prime}$ at radius $1$. Moreover, for any
$R>2$ and $i>>1$, each $V_{i}^{\prime}\llcorner(A_{R,1},g_{i}^{\prime})$ is a
$(\mathbf{C},m,\beta)$-strong-cone region. Finally, note that we continue to
have $C^{3}$ convergence $g_{i}^{\prime}\to g_{\mathrm{eucl}}$ on compact
subsets of $\mathbb{R}^{8}$.

Monotonicity (18) and the fact $a_{i}\to 0$ imply that
$\theta_{V_{i}^{\prime}}(0,R)\leq(m+o(1))\theta_{\mathbf{C}}(0)$ for any
particular $R>0$, where here $o(1)\to 0$ as $i\to\infty$. Trivially we also
have $\mathrm{index}(V_{i}^{\prime},B_{R},g_{i}^{\prime})\leq I$ for every
$R>0$. Therefore, by Lemma 4.4, we can find an
$V^{\prime}\in\mathcal{ISV}_{7}(\mathbb{R}^{8},g_{\mathrm{eucl}})$ with
$\theta_{V^{\prime}}(0,\infty)\leq m\theta_{\mathbf{C}}(0)$, and at most $I$
points $\mathcal{I}^{\prime}\subset\mathrm{spt}V^{\prime}$, so that
$V_{i}^{\prime}\to V^{\prime}$ as varifolds in $\mathbb{R}^{8}$ and in $C^{2}$
on compact subsets of
$\mathbb{R}^{8}\setminus(\mathrm{sing}V^{\prime}\cup\mathcal{I}^{\prime})$.
Provided $\beta(l)$ is sufficiently small, Arzela-Ascoli, Theorem 5.1, and
(14) imply that $V^{\prime}\llcorner(A_{\infty,1},g_{\mathrm{eucl}})$ is a
$(\mathbf{C},m,\beta)$-strong-cone region, and $V_{i}^{\prime}\to V^{\prime}$
in $C^{2}$ on compact subsets of $\mathbb{R}^{8}\setminus\overline{B_{1/8}}$.
In particular
$\mathrm{sing}V^{\prime}\cup\mathcal{I}^{\prime}\subset\overline{B_{1/8}}$ is
a finite set.

I claim that any tangent cone to $V^{\prime}$ at infinity takes the form
$m[\mathbf{C}^{\prime}]$ for some
$\mathbf{C}^{\prime}\in\mathcal{C}(\mathbf{C})$. To see this, take any
$R_{i}\to\infty$, and suppose $(\eta_{0,R_{i}})_{\sharp}V^{\prime}\to
V^{\prime\prime}\in\mathcal{ISV}_{7}(\mathbb{R}^{8})$ as varifolds. Since (by
Lemma 4.4)
$\mathrm{index}(V^{\prime\prime},\mathbb{R}^{8},g_{\mathrm{eucl}})<\infty$ and
$V^{\prime\prime}$ is a cone, $V^{\prime\prime}$ is stable and hence
$V^{\prime\prime}=m^{\prime\prime}[\mathbf{C}^{\prime\prime}]$ for some