# Extensions of Schoen–Simon–Yau and Schoen–Simon theorems via iteration à la
De Giorgi

Costante Bellettini
University College London

###### Abstract

We give an alternative proof of the Schoen–Simon–Yau curvature estimates and
associated Bernstein-type theorems (1975), and extend the original result by
including the case of $6$-dimensional (stable minimal) immersions. The key
step is an $\mathop{\varepsilon}$-regularity theorem, that assumes smallness
of the scale-invariant $L^{2}$ norm of the second fundamental form.

Further, we obtain a graph description, in the Lipschitz multi-valued sense,
for any stable minimal immersion of dimension $n\geq 2$, that may have a
singular set $\Sigma$ of locally finite $\mathcal{H}^{n-2}$-measure, and that
is weakly close to a hyperplane. (In fact, if $\mathcal{H}^{n-2}(\Sigma)=0$,
the conclusion is strengthened to a union of smooth graphs.) This follows
directly from an $\mathop{\varepsilon}$-regularity theorem, that assumes
smallness of the scale-invariant $L^{2}$ tilt-excess (verified when the
hypersurface is weakly close to a hyperplane). A further direct consequence is
that (for an immersed hypersurface of the type considered) any tangent cone
supported on a hyperplane is the unique tangent. Specialising the multi-valued
decomposition to the case of embeddings, we recover the Schoen–Simon theorem
(1981).

In both $\mathop{\varepsilon}$-regularity theorems the relevant quantity
(respectively, length of the second fundamental form and tilt function) solves
a non-linear PDE on the immersed minimal hypersurface. The proof is carried
out intrinsically (without linearising the PDE) by implementing an iteration
method à la De Giorgi (from the linear De Giorgi–Nash–Moser theory). Stability
implies estimates (intrinsic weak Caccioppoli inequalities) that make the
iteration effective despite the non-linear framework. (In both
$\mathop{\varepsilon}$-regularity theorems the method gives explicit constants
that quantify the required smallness.)

###  1 Introduction

Part I: curvature estimates. In the renowned 1975 work, Schoen–Simon–Yau
proved that any properly immersed two-sided stable minimal hypersurface $M$ in
$\mathbb{R}^{n+1}$, with $n\leq 5$, and with Euclidean mass growth at
infinity, is necessarily a union of affine hyperplanes. We develop an
alternative approach to this, which also covers the case $n=6$ that had
remained open. More precisely, we prove:

######  Theorem 1 (Bernstein-type theorem).

Let $M$ be a (smooth) properly immersed two-sided stable minimal hypersurface
in $\mathbb{R}^{n+1}$, for $n\in\\{2,3,4,5,6\\}$, with Euclidean mass growth
at infinity, i.e. there exists $\Lambda\in(0,\infty)$ such that
$\mathcal{H}^{n}\big{(}M\cap B_{R}^{n+1}(0)\big{)}\leq\Lambda R^{n}$ for all
$R>0$. Then $M$ is a union of affine hyperplanes.

As is well-known, an equivalent formulation of this property is given via a
priori (interior) curvature estimates, as follows:

######  Theorem 2 (Pointwise curvature estimates).

Assume that $M$ is a (smooth) properly immersed two-sided stable minimal
hypersurface in $B_{4R}^{n+1}(0)$, with $0\in M$ and $n\in\\{2,3,4,5,6\\}$,
with $\frac{\mathcal{H}^{n}\big{(}M\cap
B_{4R}^{n+1}(0)\big{)}}{(4R)^{n}}\leq\Lambda\in(0,\infty)$. There exists
$\beta>0$ depending only on $\Lambda$ and $n$ such that

$\sup_{x\in B_{\frac{R}{2}}^{n+1}(0)}|A_{M}|(x)\leq\frac{\beta}{R},$

where $A_{M}$ is the second fundamental form of $M$.

######  Remark 1.1.

As usual with interior-type estimates, the choice of $\frac{1}{8}$ as ratio
between the relevant radii is arbitrary. Any ratio smaller than $1$ can be
allowed, and the constant $\beta$ would depend on the chosen ratio.

As recalled above, the Schoen–Simon–Yau theory obtained these theorems for
$n\leq 5$, see [14, Theorem 3]. For $n=6$, the validity of these properties
under an additional multiplicity-$2$ condition, more precisely under the
restriction $\Lambda<3\,\omega_{6}$, was obtained by Wickramasekera, see [18,
Theorems 9.1 and 9.2] (the notation $\omega_{n}$ stands for the $n$-volume of
the $n$-dimensional unit ball). If $M$ is assumed to be properly embedded,
rather than immersed, the above results are known to be valid for $n\leq 6$ in
view of the fundamental sheeting theorem by Schoen–Simon, see [13, Theorems 1
and 3]. For $n\geq 7$, on the other hand, it has long been known that the
situation is drastically different and the above results do not hold, even
assuming that $M$ is properly embedded, as in the example of the Hardt–Simon
foliation [9].

We recall that the case $n=2$ can be treated by means of a logarithmic cut-off
argument, and, in fact, it has long been known that the above theorems hold
for $n=2$ without any mass hypothesis (see do Carmo–Peng [6], Fischer-
Colbrie–Schoen [8], and Pogorelov [11]). Very recently, the Euclidean mass
growth assumption has been shown to be redundant for $n=3$ in the work of
Chodosh–Li [3], which resolved a long-standing conjecture of Schoen (see also
subsequent alternative proofs by the same authors [4] and by
Catino–Mastrolia–Roncoroni [2]).

We obtain Theorems 1 and 2 as a consequence of an
$\mathop{\varepsilon}$-regularity theorem, in which the relevant (small)
quantity is the scale-invariant $L^{2}$-norm of the second fundamental form:

######  Theorem 3 ($\mathop{\varepsilon}$-regularity for the second
fundamental form).

Let $n\leq 6$. There exists $\epsilon_{0}>0$, depending only $n$ (a
sufficiently small $\epsilon_{0}$ is explicitly given in (12) below) with the
following significance. Let $M$ be a properly immersed two-sided stable
minimal hypersurface in $B_{2R}^{n+1}(0)$, with $0\in M$ and with

$\frac{1}{(2R)^{n-2}}\int_{M\cap B_{2R}^{n+1}(0)}|A_{M}|^{2}\leq\epsilon_{0}.$

Then for every $x\in M\cap B_{R/2}^{n+1}(0)$ we have
$|A_{M}|(x)\leq\frac{1}{R}$. More precisely, in the above smallness regime, we
have, for a (explicit) dimensional constant $c(n)$,

$\sup_{M\cap
B^{n+1}_{R/2}(0)}|A_{M}|\leq\frac{c(n)}{R}\Bigg{(}\frac{1}{(2R)^{n-2}}\int_{M\cap
B_{2R}^{n+1}(0)}|A_{M}|^{2}\Bigg{)}^{\frac{1}{2}}.$

Theorem 3 is established by PDE methods, working intrinsically on the immersed
hypersurface. The relevant (non-linear) PDE is the Simons equation for
$|A_{M}|$. The proof is obtained by implementing an iteration scheme à la De
Giorgi, in the style of the linear theory in [5] (the widely known De
Giorgi–Nash–Moser theory). The iteration relies on the validity of a weak
intrinsic Caccioppoli inequality, valid for level set truncations of
$|A_{M}|$. We establish this inequality in Lemma 2.1, from the associated PDE,
making (essential) use of the stability hypothesis to control the terms that
escape the linear PDE theory framework.

Theorems 1 and 2 then follow employing soft classical geometric measure theory
arguments: Allard’s compactness, tangent cone analysis, Federer’s dimension
reduction. Ultimately as a consequence of the well-known Simons classification
of stable cones ([17]), the analysis only needs to address two scenarios, one
in which $M$ is sufficiently close to a hyperplane with multiplicity and a
second in which $M$ is sufficiently close to a classical cone.111A classical
cone is the union of three or more (distinct) closed half-hyperplanes, having
for boundary a common $(n-1)$-dimensional subspace of $\mathbb{R}^{n+1}$, and
all intersecting at said boundary, each half-hyperplane endowed with an
integer multiplicity. The common $(n-1)$-dimensional subspace is also referred
to as the spine, a term which in general denotes the maximal subspace along
which a cone is translation invariant. (Closeness is understood in the sense
of varifolds.) In both scenarios, the smallness condition of Theorem 3 is
verified. In the first one, this is a consequence of the control of the tilt-
excess by the height-excess (recalled in Remark 4.2) and of Schoen’s
inequality ([12], [13], see also (16) below taken with $k=0$). In the second
scenario, the smallness is a consequence of the conclusions obtained in the
first scenario, applied away from the spine of the classical cone, and of a
higher integrability estimate used close to the spine.

######  Remark 1.2.

For $n=3$, the proof of Theorem 3 leads to the conclusion without using any
smallness assumption, thus it establishes directly Theorems 1 and 2. In fact,
it yields the case $n=3$ of Theorem 2 with $\beta\leq c\Lambda$ for an
explicit constant $c$. Combining this with [4], one obtains an absolute
explicit constant $D$ (independent of $\Lambda$) such that
$\sup_{B_{\frac{R}{4}}}|A|\leq\frac{D}{R}$ in the hypotheses of Theorem 2 for
$n=3$ (see Corollary 3 and Remark A.2).

Part II: Towards a compactness theory for branched stable minimal immersions.
In the second part of this work, we consider arbitrary dimensions and allow
stable minimal immersed hypersurfaces to have a singular set. More precisely,
we enlarge the class of two-sided stable minimal immersions by allowing $M$ to
have a singular set with vanishing $2$-capacity, in particular, we allow it to
have locally finite $\mathcal{H}^{n-2}$-measure.

The key result here is an $\mathop{\varepsilon}$-regularity result for the
scale-invariant tilt excess, Theorem 4 below, valid in all dimensions $n\geq
2$. The tilt function on $M$ is

$g=\sqrt{1-(\nu_{M}\cdot e_{n+1})^{2}},$

where $\nu_{M}$ is a choice of unit normal to $M$ and $e_{n+1}$ is (any fixed
unit vector, which we can without loss of generality assume to be) the last
coordinate vector. The tilt-function thus varies in $[0,1]$, with values being
higher where the tangent to $M$ tilts more with respect to the reference
hyperplane $\\{x_{n+1}=0\\}$. The smallness condition is assumed on the scale-
invariant $L^{2}$-norm of $g$. More precisely, letting $C_{r}$ denote the
cylinder $B_{r}^{n}(0)\times(-r,r)$, for $R>0$ the scale-invariant $L^{2}$
tilt-excess of $M$ on $C_{R}$ is the quantity $E_{M}(R)$ defined by

$E_{M}(R)^{2}=\frac{1}{R^{n}}\int_{M\cap C_{R}}(1-(\nu_{M}\cdot
e_{n+1})^{2}).$

We have:

######  Theorem 4 ($\mathop{\varepsilon}$-regularity for the tilt).

Let $n\geq 2$. Let $M$ be a properly immersed, two-sided, stable minimal
hypersurface in $C_{2R}\setminus\Sigma$, where $\Sigma$ is closed in
$C_{2R}=B^{n}_{2R}(0)\times(-2R,2R)$, and with $\text{cap}_{2}(\Sigma)=0$ (in
particular, $\mathcal{H}^{n-2}\big{(}\Sigma\cap K\big{)}<\infty$ for every
$K\subset\subset C_{2R}$ is permitted). There exists a positive dimensional
constant $k(n)$ (a sufficiently small $k(n)$ is given explicitly in (23)
below) with the following significance. Assume that

$E_{M}(R)^{2}=\frac{1}{R^{n}}\int_{M\cap C_{R}}g^{2}\leq k(n).$

Then

$\sup_{M\cap C_{\frac{R}{2}}}g\leq\frac{1}{2n}.$

In fact, there exists a dimensional constant $c(n)$ such that, if
$E_{M}(R)^{2}\leq k(n)$, then for every $x\in M\cap C_{\frac{R}{2}}$ we have

$g(x)\leq c(n)\big{(}E_{M}(R)\big{)}^{\frac{4}{4+n}}.$

Theorem 4 is applicable when $M\cap C_{R}$ is sufficiently close to a
hyperplane, which we can assume to be $\\{x_{n+1}=0\\}$ by a suitable
rotation. This follows from the standard control of $E_{R}(M)$ by means of the
$L^{2}$ height-excess (as recalled in Remark 4.2 this is an easy consequence
of the minimality assumption). The bound on $g$ obtained in Theorem 4 forces a
decomposition of $\overline{M}\cap C_{\frac{R}{2}}$ into a union of graphs
(over $B^{n+1}_{\frac{R}{2}}\cap\\{x_{n+1}=0\\}$), where $\overline{M}$
denotes the closure of $M$ in $C_{2R}$. In the general case considered in
Theorem 4 these graphs are Lipschitz. However, with stronger assumptions we
obtain stronger conclusions as well. We illustrate three main instances: the
general case of singular immersions, the case of smooth immersions, the case
of singular embeddings, respectively Theorems 5, 6 and 7 below. All three
follow from the $\mathop{\varepsilon}$-regularity result, Theorem 4, very
directly.

Theorem 4 is (as was the case for Theorem 3) established by PDE methods,
working intrinsically on the immersed hypersurface $M$.

The relevant (non-linear) PDE, for the tilt-function $g$, is a direct
consequence of the Jacobi field equation for $(\nu\cdot e_{n+1})$. We prove
that a weak intrinsic Caccioppoli inequality is valid for level set
truncations of $g$, see Lemma 4.2; this uses the associated PDE and the
stability hypothesis (to control the terms that escape the linear PDE theory
framework, which involve $|A_{M}|$). We then implement an iteration à la De
Giorgi. In Remarks 2.4 and 4.4 we discuss similarities and differences between
the two intrinsic weak Caccioppoli inequalities (Lemmas 2.1 and 4.2), as well
as compare them to those in De Giorgi’s work [5].

######  Theorem 5 (sheeting theorem for singular immersions).

In the hypotheses of Theorem 4, with the further assumption that $\sup_{M\cap
C_{2R}}|x_{n+1}|<\frac{R}{2}$, we have

$\overline{M}\cap C_{\frac{R}{2}}=\cup_{j=1}^{q}\text{graph}(u_{j})$

for some $q\in\mathbb{N}$, and $u_{j}:B^{n}_{\frac{R}{2}}\to\mathbb{R}$ are
Lipschitz functions, with Lipschitz constant at most $\frac{1}{2n}$ and
$u_{j}\leq u_{j+1}$ for every $j\in\\{1,\ldots,q-1\\}$. (We identify
$B^{n}_{\frac{R}{2}}$ with $B^{n+1}_{\frac{R}{2}}(0)\cap\\{x_{n+1}=0\\}$ and
the target $\mathbb{R}$ with the $x_{n+1}$ coordinate axis.) More precisely,
the Lipschitz constant of each $u_{j}$ is bounded by
$c(n)(E_{M}(R))^{\frac{n}{n+4}}$, with $c(n)$ a dimensional constant.

In fact, $M\cap C_{\frac{R}{2}}$ in Theorem 5 is naturally a smooth $q$-valued
function on $B^{n}_{\frac{R}{2}}\setminus\pi(\Sigma)$, where
$\pi:\mathbb{R}^{n}\times\mathbb{R}\to\mathbb{R}^{n}$ is the standard
projection. In general, we do not have $q$ smooth graphs on
$B_{\frac{R}{2}}\setminus\pi(\Sigma)$: the size of $\Sigma$ permits classical
branching, therefore smoothness only holds for the $q$-valued function and, in
general, there is no “selection” of $q$ smooth functions on
$B_{\frac{R}{2}}\setminus\pi(\Sigma)$. On the other hand, one can easily write
$\overline{M}\cap C_{\frac{R}{2}}$ as the union of graphs of $q$ Lipschitz
functions by ordering the $q$ values increasingly and extending the Lipschitz
functions across $\pi(\Sigma)$, as done in the statement of Theorem 5.

The multi-valued graph structure obtained in Theorem 5 rules out, for example,
that (in a branched stable minimal immersion with singular set of locally
finite $(n-2)$-measure) there may be an accumulation of necks (connecting
different sheets) onto a flat branch point.

######  Remark 1.3.

Working with immersions of the same type as in Theorem 5, and under an
additional ‘multiplicity 2’ assumption, Wickramasekera [18] obtained that when
the $L^{2}$ height-excess $\int_{C_{R}}|x_{n+1}|^{2}$ is sufficiently small
then a sheeting description is valid by means of a $C^{1,\alpha}$ $2$-valued
function. (The strategy in [18] involves a $2$-valued Lipschitz approximation
of $M$, followed by a linearisation of the problem, which in particular
prevents a quantitative smallness condition.)

Theorem 5 advances towards a compactness theory for branched stable minimal
immersions. (This was obtained for a “multiplicity $2$ class” in [18].) The
natural missing step is the analysis of the situation in which, rather than
being close to a hyperplane with multiplicity, $M$ is close to a classical
cone. In view of Theorem 5, and of the multiplicity-$2$ case in [18], it seems
natural to expect that:

_Conjecture_ : the class of branched two-sided stable minimal $n$-dimensional
immersions with singular set of locally finite $(n-2)$-measure is compact
under varifold convergence.

A further natural aim would then be to obtain a finer structure result for
said singular set, in the style of [18, Theorem 1.5]. It may be possible to
use an intrinsic approach. This lies outside the scope of this work.

######  Remark 1.4 (_Unique tangent hyperplanes_).

An immediate byproduct of Theorem 5, for the class of immersions under study,
is that: if $x\in\overline{M}$ is such that one tangent cone (in the sense of
varifolds) to $M$ at $x$ is supported on a hyperplane, then that is the unique
tangent cone at $x$ (Corollary 1). Similarly, assuming that $M$ is entire,
with Euclidean mass growth, if one tangent cone at infinity is a hyperplane
with multiplicity, then it is the unique tangent at infinity (Corollary 2).

If the singular set $\Sigma$ in Theorem 4 is a priori assumed empty then the
graphical decomposition is stronger and prompts a linear PDE behaviour, with a
linear (interior) control of $\sup|A_{M}|$ by $E_{M}(R)$:

######  Theorem 6 (sheeting theorem for smooth immersions).

In the hypotheses of Theorem 4, with the further assumptions that $\sup_{M\cap
C_{2R}}|x_{n+1}|<\frac{R}{2}$ and that $\Sigma=\emptyset$ (that is, $M$ is a
closed immersed hypersurface in $C_{2R}$), we have

$M\cap C_{\frac{R}{2}}=\cup_{j=1}^{q}\text{graph}(v_{j}),$

where $v_{j}:B^{n}_{\frac{R}{2}}(0)\equiv
B_{\frac{R}{2}}^{n+1}(0)\cap\\{x_{n+1}=0\\}\to\mathbb{R}\equiv\text{span}(e_{n+1})$
are smooth functions and $q\in\mathbb{N}$. (We note that these graphs are not
ordered, they may cross.) Moreover, $\sup_{j\in\\{1,\ldots,q\\}}\|\nabla
v_{j}\|_{C^{1,\alpha}\big{(}B^{n}_{\frac{R}{2}}(0)\big{)}}\leq c(n)E_{M}(R)$,
for a dimensional constant $c(n)$. In particular,

$\sup_{C_{\frac{R}{2}}\cap M}|A_{M}|\leq c(n)E_{M}(R).$

######  Remark 1.5.

Theorem 6 rules out the appearance of a flat branch point when taking a
(varifold) limit of smooth stable minimal immersions.

######  Remark 1.6 (_Sufficiency of $\mathcal{H}^{n-2}(\Sigma)=0$_).

The assumption $\Sigma=\emptyset$ in Theorem 6 can be weakened to
$\mathcal{H}^{n-2}(\Sigma)=0$. Indeed, under this assumption,
$B_{\frac{R}{2}}\setminus\pi(\Sigma)$ is simply connected, which implies
(using the conclusions of Theorem 5) that $M$ can be written as the union of
graphs (not ordered ones) of smooth functions on
$B_{\frac{R}{2}}\setminus\pi(\Sigma)$. Then a removal of singularity for the
minimal surface PDE ([15]) shows that in fact $\Sigma=\emptyset$. In view of
this (in analogy with the earlier discussion on the branched case), a
compactness theory for stable minimal immersions with a codimension-$7$
singular set could be obtained upon addressing the natural missing step, in
which $M$ is close to a classical cone, rather than to a hyperplane with
multiplicity. (Again, [18, Theorem 1.3] obtained this for a “multiplicity $2$
class”.)

If we instead specialise Theorem 5 to hypersurfaces $M$ that are _embedded_
away from $\Sigma$, then by removal of singularities for the minimal surface
PDE, used for each function $u_{j}$, we recover (a quantitative version of)
the well-known Schoen–Simon sheeting theorem ([13, Theorem 1]):

######  Theorem 7 (sheeting theorem for singular embeddings).

Let $n\geq 2$. Let $M$ be a properly embedded, two-sided stable minimal
hypersurface in $C_{2R}\setminus\Sigma$, where $\Sigma$ is closed in
$C_{2R}=B^{n}_{R}(0)\times(-R,R)$, with locally finite
$\mathcal{H}^{n-2}$-measure, or, more generally, with
$\text{cap}_{2}(\Sigma)=0$. Assume that $\sup_{M\cap
C_{2R}}|x_{n+1}|<\frac{R}{2}$ and

$E_{M}(R)^{2}=\frac{1}{R^{n}}\int_{M\cap C_{R}}(1-\nu\cdot e_{n+1})^{2}\leq
k(n),$

where $k(n)$ is the (positive) dimensional constant in (23) and $\nu$ is a
choice of unit normal to $M$. Then

$\overline{M}\cap C_{\frac{R}{2}}=\cup_{j=1}^{q}\text{graph}(u_{j})$

with $u_{j}:B_{\frac{R}{2}}\to\mathbb{R}$ smooth, $u_{j}<u_{j+1}$ for every
$j$. In particular, $\overline{M}\cap C_{\frac{R}{2}}$ is smoothly embedded
(equivalently, $M$ extends smoothly across $\Sigma$ in $C_{\frac{R}{2}}$), and
$\sup_{C_{\frac{R}{2}}\cap M}|A_{M}|\leq c(n)E_{M}(R)$ for a dimensional
constant $c(n)$.

We recall that Theorem 7 leads (by fairly standard arguments) to the renowned
compactness and regularity theory [13, Theorems 2 and 3] for stable minimal
embedded hypersurfaces that are allowed to possess a singular set of locally
finite $\mathcal{H}^{n-2}$-measure (a posteriori, the singular set has
dimension at most $n-7$, and is discrete in the case $n=7$, and empty for
$n\leq 6$).

We thus obtain an alternative and hopefully more immediate route to (the main
component of) the Schoen–Simon theory. (The approach in [13] involves a
partial $q$-valued graph decomposition of the embedding, and a linearisation
of the problem, both of which we avoid.)

The impact of Schoen–Simon’s compactness, and of Schoen–Simon–Yau’s curvature
estimates, for developments in analysis and geometry over the last half
century, cannot be overstated.

##  Part I Curvature estimates

We will denote by $M$ (and by $M_{\ell}$, $\ell\in\mathbb{N}$, when
considering a sequence) a smooth two-sided properly immersed stable minimal
hypersurface in an open set $U\subset\mathbb{R}^{n+1}$. Typically, the open
set $U$ will be a ball $B_{R}^{n+1}(0)$, or the whole of $\mathbb{R}^{n+1}$,
or a cylinder of the form $B_{R}^{n}(0)\times(-R,R)$. In other words
$M=\iota(S)$, with $S$ an $n$-dimensional manifold and $\iota:S\to U$ a proper
two-sided stable minimal immersion. We recall that the stability condition is
the non-negativity of the second variation of the $n$-area, and that this
amounts to the validity of

$\int_{S}|A_{M}|^{2}\phi^{2}\leq\int_{S}|\nabla\phi|^{2}$

for any $\phi\in C^{1}_{c}(S)$, where $S$ is endowed with the pull-back metric
from $U$, $\nabla$ is the metric gradient on $S$ and $|A_{M}|$ the length of
the second fundamental form. We note that whenever $\varphi\in C^{1}_{c}(U)$,
then $\varphi\circ\iota\in C^{1}_{c}(S)$ (since the immersion is proper); with
a slight abuse of notation, we will write the integrals directly on $M$, with
$\int_{M}|\nabla\varphi|^{2}$ in place of
$\int_{S}|\nabla(\varphi\circ\iota)|^{2}$ and the inequality taking the form
$\int_{M}|A_{M}|^{2}\varphi^{2}\leq\int_{M}|\nabla\varphi|^{2}$.

###  2 Proof of Theorem 3

In this section we prove Theorem 3. We recall the well-known Simons identity
([17]), for the second fundamental form $A$ of a minimal hypersurface:

$\frac{1}{2}\Delta|A|^{2}=|\nabla A|^{2}-|A|^{4}.$ (1)

Clearly, $|\nabla A|\geq\big{|}\nabla|A|\big{|}$; in [14, (1.33)] it is shown
that the minimality condition implies the following improved inequality, with
$c=\frac{2}{n}$:

$|\nabla A|^{2}\geq(1+c)\big{|}\nabla|A|\big{|}^{2}.$ (2)

We will also use the following variant of (1): as
$\frac{1}{2}\Delta|A|^{2}=|A|\Delta|A|+|\nabla|A||^{2}$, we find

$|A|\Delta|A|=|\nabla A|^{2}-\big{|}\nabla|A|\big{|}^{2}-|A|^{4}.$ (3)

The following lemma contains the relevant weak (intrinsic) Caccioppoli
inequality for the level set truncations of $|A|$:

######  Lemma 2.1.

Let $M$ be a properly immersed smooth two-sided stable minimal hypersurface in
$U\subset\mathbb{R}^{n+1}$. For any $k\geq 0$ and any $\eta\in C^{1}_{c}(U)$
we have

$\displaystyle\int_{\\{|A|>k\\}}\Bigg{(}1-\frac{k}{|A|}\Bigg{)}\,\big{|}\nabla|A|\big{|}^{2}\,\eta^{2}\leq\frac{1}{c}\int((|A|-k)^{+})^{2}|\nabla\eta|^{2}+$
$\displaystyle\frac{k}{c}\int((|A|-k)^{+})^{3}\,\eta^{2}+\frac{2k^{2}}{c}\int((|A|-k)^{+})^{2}\,\eta^{2}+\frac{k^{3}}{c}\int(|A|-k)^{+}\,\eta^{2}.$

######  Remark 2.1.

As mentioned above, the integrals are implicitly understood to be on $S$, with
$\eta\circ\iota$ in place of $\eta$. For $k=0$ the inequality is
$\int\big{|}\nabla|A|\big{|}^{2}\,\eta^{2}\leq\frac{1}{c}\int|A|^{2}|\nabla\eta|^{2}$,
which appears as an intermediate step along the proof of [14, Theorem 1].

###### Proof.

We use the stability inequality with the Lipschitz test function
$(|A|-k)^{+}\eta$, for $k\in[0,\infty)$ and $\eta\in C^{1}_{c}(U)$. We note
that ${(|A|-k)^{+}}^{2}\in C^{1}(M)\cap W^{2,\infty}_{\text{loc}}(M)$: indeed,
being Lipschitz, its (distributional) gradient is the function
$\nabla{(|A|-k)^{+}}^{2}=2{(|A|-k)^{+}}\nabla{(|A|-k)^{+}}=2{(|A|-k)^{+}}\nabla|A|$,
which in turn is locally Lipschitz. We find

$\displaystyle\int|A|^{2}(|A|-k)^{+})^{2}\eta^{2}\leq\int|\nabla((|A|-k)^{+}\eta)|^{2}=$
$\displaystyle\int|\nabla(|A|-k)^{+}|^{2}\eta^{2}+2\int(|A|-k)^{+}\,\eta\,\nabla(|A|-k)^{+}\,\nabla\eta+\int((|A|-k)^{+})^{2}|\nabla\eta|^{2}$
$\displaystyle=\int|\nabla(|A|-k)^{+}|^{2}\eta^{2}+\frac{1}{2}\underbrace{\int\nabla((|A|-k)^{+})^{2}\,\nabla\eta^{2}}+\int((|A|-k)^{+})^{2}|\nabla\eta|^{2}=$

and integrating by parts the braced term we can continue the equality chain

$\displaystyle=\int|\nabla(|A|-k)^{+}|^{2}\eta^{2}-\frac{1}{2}\int_{\\{|A|>k\\}}\Delta|A|^{2}\,\eta^{2}+\int_{\\{|A|>k\\}}\frac{k}{|A|}|A|\Delta|A|\,\eta^{2}$
$\displaystyle+\int((|A|-k)^{+})^{2}|\nabla\eta|^{2}$
$\displaystyle\underbrace{=}_{(\ref{eq:Simons}),(\ref{eq:Simons2})}\int_{\\{|A|>k\\}}\big{|}\nabla|A|\big{|}^{2}\eta^{2}+\int_{\\{|A|>k\\}}(-|\nabla
A|^{2}+|A|^{4})\,\eta^{2}+$
$\displaystyle\int_{\\{|A|>k\\}}\frac{k}{|A|}\big{(}|\nabla
A|^{2}-\big{|}\nabla|A|\big{|}^{2}\big{)}\,\eta^{2}-k\int_{\\{|A|>k\\}}|A|^{3}\,\eta^{2}+\int((|A|-k)^{+})^{2}|\nabla\eta|^{2}.$

The left-most side (of the above chain of inequalities) is expanded as
follows:
$\int|A|^{2}(|A|-k)^{+})^{2}\eta^{2}=\int_{\\{|A|>k\\}}(|A|^{4}-2k|A|^{3}+k^{2}|A|^{2})\eta^{2}$.
We thus find

$\displaystyle\int_{\\{|A|>k\\}}\left(1-\frac{k}{|A|}\right)(|\nabla
A|^{2}-\big{|}\nabla|A|\big{|}^{2})\eta^{2}\leq$
$\displaystyle\int((|A|-k)^{+})^{2}|\nabla\eta|^{2}+k\int_{\\{|A|>k\\}}|A|^{3}\,\eta^{2}-k^{2}\int_{\\{|A|>k\\}}|A|^{2}\,\eta^{2}$

and using (2) we conclude

$\int_{\\{|A|>k\\}}\Bigg{(}1-\frac{k}{|A|}\Bigg{)}\,\big{|}\nabla|A|\big{|}^{2}\,\eta^{2}\leq$
(4)
$\frac{1}{c}\int\big{(}(|A|-k)^{+}\big{)}^{2}|\nabla\eta|^{2}+\frac{k}{c}\int|A|^{2}(|A|-k)^{+}\,\eta^{2}.$

The desired inequality follows upon rewriting the last term on the right-hand-
side of (4), on the set ${\\{|A|>k\\}}$:

$|A|^{2}(|A|-k)=\left(|A|-k+k\right)^{2}(|A|-k)=(|A|-k)^{3}+2k(|A|-k)^{2}+k^{2}(|A|-k).$
(5)

∎

Lemma 2.1 provides the weak intrinsic Caccioppoli inequality that will lead,
through an iteration scheme à la De Giorgi, to Theorem 3.

######  Remark 2.2.

Note that both assumption and conclusion in Theorem 3 are scale-invariant. The
scale-invariant quantity $\frac{1}{(R)^{n-2}}\int_{M\cap B_{R}(0)}|A|^{2}$ is
uniformly bounded under the Euclidean mass growth hypothesis that we have.
Indeed, stability used with a test function $\varphi\in C^{1}_{c}(B_{2R}(0))$
with $\varphi\equiv 1$ on $B_{R}(0)$ and $|\nabla\varphi|\leq\frac{2}{R}$
gives
$\int_{B_{R}(0)}|A|^{2}\leq\frac{4}{R^{2}}\Big{(}\int_{B_{2R}(0)}1\Big{)}\leq
2^{n+2}R^{n-2}\Lambda$.

######  Remark 2.3.

The standard catenoids for $n\geq 3$, for which
$\frac{1}{R^{n-2}}\int_{B_{R}(0)\cap M}|A|^{2}\to 0$ as $R\to\infty$, show
that stability is essential for Theorem 3. (On the other hand, the dimensional
restriction may not be essential.) Catenoids also show that, given $M$
minimal, the relevant “scale-invariant energy”
$\frac{1}{R^{n-2}}\int_{B_{R}(0)\cap M}|A|^{2}$ is not an increasing function
of $R$ (unlike in several well-known $\mathop{\varepsilon}$-regularity
theorems).

###### Proof of Theorem 3.

We will assume $n\geq 3$ (see Remark 2.5) and consider the sequences
$k_{\ell}=d-\frac{d}{2^{\ell-1}}$ and
$R_{\ell}=\frac{R}{2}+\frac{R}{2^{\ell}}$ for $\ell\in\\{1,2,\ldots,\\}$
(respectively increasing and decreasing), where $d>0$ is for the moment left
undetermined, and will be quantified in terms of
$\frac{1}{(2R)^{n-2}}\int_{B_{2R}}|A|^{2}$. Here and below, when writing
integrals on $B_{r}=B^{n+1}_{r}(0)$ we implicitly understand that the
integration is on $\iota^{-1}(B^{n+1}_{r}(0))$.

Using Lemma 2.1 with $k_{\ell}$ in place of $k$, noting the inclusion
$\\{|A|>k_{\ell}\\}\supset\\{|A|>k_{\ell+1}\\}$, and that on the set
$\\{|A|>k_{\ell+1}\\}$ we have $\left(1-\frac{k_{\ell}}{|A|}\right)^{+}\geq
1-\frac{k_{\ell}}{k_{\ell+1}}=\frac{d}{k_{\ell+1}2^{\ell}}\geq\frac{1}{2^{\ell}}$
we have

$\displaystyle\frac{1}{2^{\ell}}\int_{\\{|A|>k_{\ell+1}\\}}|\nabla|A||^{2}\,\eta^{2}\leq\frac{1}{c}\int((|A|-k_{\ell})^{+})^{2}|\nabla\eta|^{2}+\frac{k_{\ell}}{c}\int((|A|-k_{\ell})^{+})^{3}\,\eta^{2}$
$\displaystyle+\frac{2k_{\ell}^{2}}{c}\int((|A|-k_{\ell})^{+})^{2}\,\eta^{2}+\frac{k_{\ell}^{3}}{c}\int(|A|-k_{\ell})^{+}\,\eta^{2}.$

Since

$|\nabla((|A|-k_{\ell+1})^{+}\eta)|^{2}\leq
2\chi_{\\{|A|>k_{\ell+1}\\}}|\nabla|A||^{2}\,\eta^{2}+2((|A|-k_{\ell+1})^{+})^{2}|\nabla\eta|^{2},$

and $(|A|-k_{\ell+1})^{+}\leq(|A|-k_{\ell})^{+}$, it follows that

$\displaystyle\int|\nabla((|A|-k_{\ell+1})^{+}\eta)|^{2}\leq\left(\frac{2^{\ell+1}}{c}+2\right)\int((|A|-k_{\ell})^{+})^{2}|\nabla\eta|^{2}+$
(6)
$\displaystyle\frac{k_{\ell}2^{\ell+1}}{c}\int((|A|-k_{\ell})^{+})^{3}\,\eta^{2}+\frac{2k_{\ell}^{2}2^{\ell+1}}{c}\int((|A|-k_{\ell})^{+})^{2}\,\eta^{2}+$
$\displaystyle\frac{k_{\ell}^{3}2^{\ell+1}}{c}\int(|A|-k_{\ell})^{+}\,\eta^{2}.$

We will use the following Michael–Simon inequality ([10, Theorem 2.1], see
also [1]), to bound from below the left-hand-side of (6):

$\left(\int\left|(|A|-k_{\ell+1})^{+}\eta\right|^{\frac{2n}{n-2}}\right)^{\frac{n-2}{n}}\leq
C_{MS}\int|\nabla((|A|-k_{\ell+1})^{+}\eta)|^{2},$ (7)

for a dimensional constant $C_{MS}$ explicitly given by
$C_{MS}=\left(\frac{2(n-1)4^{n+1}}{(n-2)\omega_{n}^{1/n}}\right)^{2}$, where
$\omega_{n}$ is the $n$-volume of the unit ball in $\mathbb{R}^{n}$.

We define, for each $\ell$, the function $\eta_{\ell}$ to be identically $1$
on $B_{R_{\ell+1}}$, with $\text{spt}\eta_{\ell}=B_{R_{\ell}}$ and
$|\nabla\eta_{\ell}|\leq\frac{2}{|R_{\ell}-R_{\ell+1}|}=\frac{2^{\ell+2}}{R}$,
and with $0\leq\eta_{\ell}\leq 1$. From (6) and (7), making the choice
$\eta=\eta_{\ell}$, we find (using $k_{\ell}\leq d$ on the right-hand-side of
(6))

$\displaystyle\frac{1}{C_{MS}}\left(\int_{B_{R_{\ell+1}}}((|A|-k_{\ell+1})^{+})^{\frac{2n}{n-2}}\right)^{\frac{n-2}{n}}\leq$
(8)
$\displaystyle\left(\frac{2^{\ell+1}}{c}+2\right)\left(\frac{4^{\ell+2}}{R^{2}}\right)\int_{B_{R_{\ell}}}((|A|-k_{\ell})^{+})^{2}+\frac{d2^{\ell+1}}{c}\int_{B_{R_{\ell}}}((|A|-k_{\ell})^{+})^{3}$
$\displaystyle+\frac{2d^{2}2^{\ell+1}}{c}\int_{B_{R_{\ell}}}((|A|-k_{\ell})^{+})^{2}+\frac{d^{3}2^{\ell+1}}{c}\int_{B_{R_{\ell}}}(|A|-k_{\ell})^{+}.$

For the right-hand-side of (8) we use Hölder’s inequality three times, first
with $1=\frac{n-2}{n}+\frac{2}{n}$,

$\int_{B_{R_{\ell}}}((|A|-k_{\ell})^{+})^{2}\leq\left(\int_{B_{R_{\ell}}}((|A|-k_{\ell})^{+})^{\frac{2n}{n-2}}\right)^{\frac{n-2}{n}}\left(\int\chi_{\\{|A|>k_{\ell}\\}\cap
B_{R_{\ell}}}\right)^{\frac{2}{n}},$

then with $\frac{n-2}{2n}+\frac{n+2}{2n}=1$,

$\int_{B_{R_{\ell}}}(|A|-k_{\ell})^{+}\leq\left(\int_{B_{R_{\ell}}}((|A|-k_{\ell})^{+})^{\frac{2n}{n-2}}\right)^{\frac{n-2}{2n}}\left(\int\chi_{\\{|A|>k_{\ell}\\}\cap
B_{R_{\ell}}}\right)^{\frac{n+2}{2n}},$

and finally (this is possible for $n\leq 6$) with exponents
$\frac{2n}{3(n-2)}$ and $\frac{2n}{6-n}$ (with
$1=\frac{3n-6}{2n}+\frac{6-n}{2n}$), where in the case $n=6$ the two exponents
are just $1$ and $\infty$ (hence for $n=6$ the second factor on the
right–hand-side in the following inequality is just $1$)

$\int_{B_{R_{\ell}}}((|A|-k_{\ell})^{+})^{3}\leq\left(\int_{B_{R_{\ell}}}((|A|-k_{\ell})^{+})^{\frac{2n}{n-2}}\right)^{\frac{3(n-2)}{2n}}\left(\int\chi_{\\{|A|>k_{\ell}\\}\cap
B_{R_{\ell}}}\right)^{\frac{6-n}{2n}}.$

We will use the notation

$S_{\ell}=\int_{B_{R_{\ell}}}((|A|-k_{\ell})^{+})^{\frac{2n}{n-2}},$

and aim for a superlinear decay estimate for this quantity as $\ell\to\infty$.

We note that $\mathcal{H}^{n}\left(\\{|A|>k_{\ell}\\}\cap B_{R_{\ell}}\right)$
can be bounded as follows, for $\ell\geq 2$, using Markov’s inequality and the
fact that on the set $\\{|A|>k_{\ell}\\}$ we have $(|A|-k_{\ell-1})^{+}\geq
k_{\ell}-k_{\ell-1}=\frac{d}{2^{\ell-1}}$:

$\\{|A|>k_{\ell}\\}\cap
B_{R_{\ell}}\subset\left\\{((|A|-k_{\ell-1})^{+})^{\frac{2n}{n-2}}\geq\frac{d^{\frac{2n}{n-2}}}{(2^{\frac{2n}{n-2}})^{\ell-1}}\right\\}\cap
B_{R_{\ell}}\Rightarrow$ $\mathcal{H}^{n}\left(\\{|A|>k_{\ell}\\}\cap
B_{R_{{\ell}}}\right)\leq\frac{(2^{\frac{2n}{n-2}})^{\ell-1}}{d^{\frac{2n}{n-2}}}\int_{B_{R_{\ell}}}((|A|-k_{\ell-1})^{+})^{\frac{2n}{n-2}}\leq\frac{(2^{\frac{2n}{n-2}})^{\ell-1}}{d^{\frac{2n}{n-2}}}S_{\ell-1}.$
(9)

The above three instances of Hölder’s inequality, together with (9), give

$\int_{B_{R_{\ell}}}((|A|-k_{\ell})^{+})^{2}\leq
S_{\ell}^{\frac{n-2}{n}}\left(\frac{(2^{\frac{2n}{n-2}})^{\ell-1}}{d^{\frac{2n}{n-2}}}S_{\ell-1}\right)^{\frac{2}{n}},$
$\int_{B_{R_{\ell}}}(|A|-k_{\ell})^{+}\leq
S_{\ell}^{\frac{n-2}{2n}}\left(\frac{(2^{\frac{2n}{n-2}})^{\ell-1}}{d^{\frac{2n}{n-2}}}S_{\ell-1}\right)^{\frac{n+2}{2n}},$
$\int_{B_{R_{\ell}}}((|A|-k_{\ell})^{+})^{3}\leq
S_{\ell}^{\frac{3(n-2)}{2n}}\left(\frac{(2^{\frac{2n}{n-2}})^{\ell-1}}{d^{\frac{2n}{n-2}}}S_{\ell-1}\right)^{\frac{6-n}{2n}}.$

With these we bound from above the right-hand-side of (8) and obtain

$\frac{1}{C_{MS}}S_{\ell+1}^{\frac{n-2}{n}}\leq\left(\left(\frac{2^{\ell+1}}{c}+2\right)\left(\frac{4^{\ell+2}}{R^{2}}\right)+\frac{2d^{2}2^{\ell+1}}{c}\right)S_{\ell}^{\frac{n-2}{n}}\left(\frac{(2^{\frac{4}{n-2}})^{\ell-1}}{d^{\frac{4}{n-2}}}\right)S_{\ell-1}^{\frac{2}{n}}$

$+\frac{d2^{\ell+1}}{c}S_{\ell}^{\frac{3n-6}{2n}}\left(\frac{(2^{\frac{6-n}{n-2}})^{\ell-1}}{d^{\frac{6-n}{n-2}}}\right)S_{\ell-1}^{\frac{6-n}{2n}}+\frac{d^{3}2^{\ell+1}}{c}S_{\ell}^{\frac{n-2}{2n}}\left(\frac{(2^{\frac{n+2}{n-2}})^{\ell-1}}{d^{\frac{n+2}{n-2}}}\right)S_{\ell-1}^{\frac{n+2}{2n}},$

for every $\ell\geq 2$. Noting that $S_{\ell}\leq S_{\ell-1}$ by definition,
and that
$2-\frac{4}{n-2}=1-\frac{6-n}{n-2}=3-\frac{n+2}{n-2}=\frac{2(n-4)}{n-2}$, the
last inequality implies (using $n=\frac{2}{c}$)

$S_{\ell+1}^{\frac{n-2}{n}}\leq(16nC_{MS})\,C^{\ell}\left(\frac{1}{R^{2}d^{\frac{4}{n-2}}}+d^{\frac{2(n-4)}{n-2}}\right)S_{\ell-1},$

where $C$ is a dimensional constant that can be explicitly taken (using rough
estimates for the constants appearing above) to be $C=2^{\frac{3n-2}{n-2}}$.

Letting $d=\frac{1}{R}$ we find $S_{\ell+1}^{\frac{n-2}{n}}\leq
32nC_{MS}\,C^{\ell}\frac{1}{R^{\frac{2(n-4)}{n-2}}}S_{\ell-1}$ and hence
arrive at the decay relation ($\ell\geq 2$)

$S_{\ell+1}\leq(32nC_{MS})^{\frac{n}{n-2}}\frac{\tilde{C}^{\ell}}{R^{\frac{2n(n-4)}{(n-2)^{2}}}}S_{\ell-1}^{1+\frac{2}{n-2}},$

where the dimensional constant $\tilde{C}=C^{\frac{n}{n-2}}$ can be explicitly
taken to be $\tilde{C}=2^{\frac{n(3n-2)}{(n-2)^{2}}}$. This relation forces
$S_{\ell}\to 0$, as long as $S_{1}$ is sufficiently small, by Lemma B.1;
precisely, if
$S_{1}\leq\frac{R^{\frac{n(n-4)}{(n-2)}}}{(32nC_{MS})^{\frac{n}{2}}\tilde{C}^{\frac{n(n-2)}{2}}}$.
(We use the relation above with $\ell$ even; the sequence $S_{\ell}$ is
decreasing by definition, hence it suffices to prove the convergence to $0$
for the subsequence of odd indices. Setting $2j=\ell$ and $T_{j}=S_{2j-1}$ for
$j\geq 1$, we obtain the recursive relation
$T_{j+1}\leq\frac{(32nC_{MS})^{\frac{n}{n-2}}}{R^{\frac{2n(n-4)}{(n-2)^{2}}}}\tilde{C}^{2j}T_{j}^{1+\frac{2}{n-2}}$
and Lemma B.1 gives $T_{j}\to 0$ if $T_{1}=S_{1}$ is as specified above.) The
smallness assumption on $S_{1}$ can be written as follows, for the associated
scale-invariant quantity (recall that $R_{1}=R$ and $k_{1}=0$ in the
definition of $S_{1}$):

$\frac{1}{R^{n-\frac{2n}{n-2}}}\int_{B_{R}}|A|^{\frac{2n}{n-2}}\leq\frac{1}{(32nC_{MS})^{\frac{n}{2}}\tilde{C}^{\frac{n(n-2)}{2}}}=\frac{1}{(32nC_{MS})^{\frac{n}{2}}\,2^{\frac{n^{2}(3n-2)}{2(n-2)}}}.$
(10)

Under the condition (10), the convergence $S_{\ell}\to 0$ obtained implies
that $|A|\leq\frac{1}{R}$ a.e. on $M\cap B_{\frac{R}{2}}$ (by our choice
$d=1$), and thus everywhere (by smoothness of $M$).

We next check that there exists $\epsilon_{0}>0$ such that (10) is implied the
hypotheses of Theorem 3. Lemma 2.1, used with $k=0$, gives
$\int_{B_{2R}}|\nabla(|A|\psi)|^{2}\leq\big{(}\frac{2}{c}+2\big{)}\int_{B_{2R}}|A|^{2}|\nabla\psi|^{2}$
for $\psi\in C^{1}_{c}(B_{2R})$. Combining this with the Michael-Sobolev
inequality, which gives
$\left(\int_{B_{2R}}(|A|\psi)^{\frac{2n}{n-2}}\right)^{\frac{n-2}{n}}\leq
C_{MS}\int_{B_{2R}}|\nabla(|A|\psi)|^{2}$, choosing $\psi$ to be identically
$1$ in $B_{R}$ and with $|\nabla\psi|\leq\frac{2}{R}$, we obtain (using
$c=\frac{2}{n}$)

$\int_{B_{R}}|A|^{\frac{2n}{n-2}}\leq
C_{MS}^{\frac{n}{n-2}}\left(\frac{4(n+2)}{R^{2}}\int_{B_{2R}}|A|^{2}\right)^{\frac{n}{n-2}},$
(11)

that is, the following inequality holds between the two relevant scale-
invariant quantities:

$\frac{1}{R^{n-\frac{2n}{n-2}}}\int_{B_{R}}|A|^{\frac{2n}{n-2}}\leq(4(n+2)\,2^{n-2}C_{MS})^{\frac{n}{n-2}}\left(\frac{1}{(2R)^{n-2}}\int_{B_{2R}}|A|^{2}\right)^{\frac{n}{n-2}}.$

We have thus established the first conclusion of Theorem 3, that is, there
exists a dimensional $\epsilon_{0}>0$ for which $|A|\leq\frac{1}{R}$ on $M\cap
B_{\frac{R}{2}}$. Explicitly, this can be quantified from the above by
requiring
$\Big{(}\epsilon_{0}\,4(n+2)\,2^{n-2}C_{MS}\Big{)}^{\frac{n}{n-2}}\leq\frac{1}{(32nC_{MS})^{\frac{n}{2}}\,2^{\frac{n^{2}(3n-2)}{2(n-2)}}}$,
that is, one can take

$\epsilon_{0}=\frac{1}{C_{MS}^{1+\frac{(n-2)}{2}}\,4(n+2)\,2^{n-2}\,(32n)^{\frac{(n-2)}{2}}\,2^{\frac{n(3n-2)}{2}}}.$
(12)

To see the second assertion of Theorem 3, we exploit the freedom on $d$. We
choose $d=\frac{1}{mR}$ for $m\in[1,\infty)$, and the conclusion $S_{\ell}\to
0$, i.e. $|A|\leq\frac{1}{mR}$ on $B_{\frac{R}{2}}$, follows if $S_{1}$ is
suitably small. Indeed, the decay relation becomes (for $\ell\geq 2$ and for
an explicit dimensional constant $C$)

$S_{\ell+1}\leq\Bigg{(}m^{\frac{4}{n-2}}+\frac{1}{m^{\frac{2(n-4)}{n-2}}}\Bigg{)}^{\frac{n}{n-2}}\frac{C^{\ell}}{R^{\frac{2n(n-4)}{(n-2)^{2}}}}S_{\ell-1}^{1+\frac{2}{n-2}}\leq
2^{\frac{n}{n-2}}m^{\frac{4n}{(n-2)^{2}}}\frac{C^{\ell}}{R^{\frac{2n(n-4)}{(n-2)^{2}}}}S_{\ell-1}^{1+\frac{2}{n-2}}.$

Hence if
$S_{1}\leq\frac{R^{\frac{n(n-4)}{(n-2)}}}{{m^{\frac{2n}{(n-2)}}}\overline{C}}$
(for an explicit dimensional constant $\overline{C}$) then $S_{\ell}\to 0$,
that is, $|A|\leq\frac{1}{mR}$ on $M\cap B_{\frac{R}{2}}$. With the same
considerations given for the case $m=1$ above, we have that the smallness
requirement on $S_{1}$ is implied by our hypotheses, as long as
$\frac{1}{(2R)^{2}}\int_{B_{2R}}|A|^{2}$ is sufficiently small. The explicit
relations that we have obtained show, in fact, that, for a (explicit)
dimensional constant $c(n)$,

$R\sup_{M\cap B_{\frac{R}{2}}}|A|\leq
c(n)\Big{(}\int_{B_{2R}}|A|^{2}\Big{)}^{\frac{1}{2}}.$

∎

######  Remark 2.4.

De Giorgi [5] exploits the Caccioppoli inequality $\int_{\\{u>k\\}\cap
B^{n}_{\rho}(p)}|Du|^{2}\leq\frac{c(n)}{(r-\rho)^{2}}\int_{\\{u>k\\}\cap
B^{n}_{r}(p)}{(u-k)^{+}}^{2}$ (for any $k$, with $\rho<r<R$) to prove his
theorem, for a weak solution $u:B^{n}_{R}(p)\to\mathbb{R}$ of a linear PDE in
divergence form with $L^{\infty}$ strictly elliptic coefficients. In Lemma 2.1
we have an _intrinsic_ Caccioppoli inequality on $M$, when $k=0$. The
multiplicative factor $\big{(}1-\frac{k}{|A|}\big{)}^{+}$ appears on the left-
hand-side for $k>0$: as shown, this does not disturb the iterative scheme.
Extra terms (that weaken the inequality further, when comparing to the
classical case) appear on the right-hand-side when $k>0$. These terms involve
$L^{p}$-norms of the truncations up to $p=3$, with multiplicative factors
$k^{4-p}$, which influence the dependence on $d$ in the decay relation. The
smallness requirement in Theorem 3 is due222As observed explicitly in Appendix
A, smallness is only needed for $n\in\\{4,5,6\\}$. to these extra terms (which
also force the dimensional bound $n\leq 6$).

While in [5] the classical Caccioppoli inequality for $(u-k)^{+}$ follows
thanks to the linearity of the PDE, in our case stability provides sufficient
control on the non-linearity of (1), leading to the weak intrinsic Caccioppoli
inequality. (In the absence of stability Theorem 3 fails, as pointed out in
Remark 2.3.)

######  Remark 2.5 ($n=2$).

We proved Theorem 3 for $n\neq 2$. The proof adapts to treat the case $n=2$ by
using the Michael–Simon inequality to embed $L^{3}$ into $W^{1,\frac{6}{5}}$
and the Hölder inequality to bound
$\int|\nabla\big{(}(|A|-{k_{\ell}})^{+}\eta_{\ell}\big{)}|^{\frac{6}{5}}$ by
means of the product
$\Big{(}\int|\nabla\big{(}(|A|-{k_{\ell}})^{+}\eta_{\ell}\big{)}|^{2}\Big{)}^{\frac{3}{5}}\mathcal{H}^{2}\Big{(}M\cap
B_{R_{\ell}}\Big{)}^{\frac{2}{5}}$. We do not carry out the iteration
explicitly, also in view of the fact that (as mentioned in the introduction)
curvature estimates for $n=2$ admit a simple treatment.

###  3 Proof of Theorems 1 and 2

The proof of Theorem 2 is essentially reduced to the analysis of two
scenarios, in both of which Theorem 3 is applicable. The first scenario
handles the case in which $M$ (properly immersed smooth two-sided and stable)
is weakly close (i.e. as a varifold) to a hyperplane with multiplicity; the
second one handles the case in which $M$ (of the same kind) is weakly close to
a classical cone. In both scenarios the conclusion is that $M$ must be a
smooth perturbation (as an immersion) of the cone in question (which is a
hyperplane with multiplicity in the former case, and has to be a union of
hyperplanes with multiplicity in the latter case).

####  3.1 Closeness to a hyperplane

Let $M$ be a properly immersed two-sided stable minimal hypersurface that is
weakly close to a hyperplane with multiplicity (as varifolds). Upon rotating
coordinates, we assume that the hyperplane is $\\{x_{n+1}=0\\}$. We recall the
following standard inequality (implied by minimality, via the first variation
formula, with a suitable choice of test function, see [16, Section 22] and
Remark 4.2 below)

$\Big{(}\frac{1}{3R}\Big{)}^{n}\int_{B_{3R}^{n+1}\cap M}|\nabla
x_{n+1}|^{2}\leq
C(n)\Big{(}\frac{2}{7R}\Big{)}^{n+2}\int_{B_{\frac{7R}{2}}^{n+1}\cap
M}|x_{n+1}|^{2},$

and $|\nabla x_{n+1}|=|\text{proj}_{TM}(e_{n+1})|=\sqrt{1-(\nu\cdot
e_{n+1})^{2}}$. This says that the $L^{2}$ height-excess controls the $L^{2}$
tilt-excess linearly (both excesses defined in a scale-invariant fashion).
Moreover, Schoen’s inequality ([12], [13, Lemma 1], see also (16) below taken
with $k=0$) gives

$\frac{1}{(2R)^{n-2}}\int_{B_{2R}^{n+1}\cap M}|A|^{2}\leq
2n\Big{(}\frac{1}{3R}\Big{)}^{n}\int_{B_{3R}^{n+1}\cap M}|\nabla
x_{n+1}|^{2},$

where the scale-invariant tilt excess appears on the right-hand-side. Here and
below, domains of the type $D\cap M$ for $D$ open, are implicitly understood
to be the inverse image of $D$ via the immersion that gives $M$ (we will use
this with $M$ belonging to a sequence of immersed hypersurfaces).

When $\hat{M}_{j}$ converge (as varifolds for $j\to\infty$) in $B_{4R}^{n+1}$
to $\\{x_{n+1}=0\\}$ with multiplicity, by the monotonicity formula we have
that $\sup_{\hat{M}_{j}\cap B_{\frac{7R}{2}}^{n+1}}|x_{n+1}|\to 0$, so we
conclude that for all sufficiently large $j$ the quantity
$\frac{1}{(2R)^{n-2}}\int_{B_{2R}^{n+1}\cap\hat{M}_{j}}|A_{\hat{M}_{j}}|^{2}$
is at most $\epsilon_{0}$ and Theorem 3 applies. As immediate consequence, we
have:

######  Lemma 3.1.

Let $\hat{M}_{j}$ be a sequence of smooth properly immersed two-sided stable
minimal hypersurfaces in $B^{n+1}_{4R}(0)$, $n\leq 6$, such that $\hat{M}_{j}$
converge (as varifolds) to $q\llbracket P\rrbracket$ as $j\to\infty$, where
$P$ is a hyperplane and $q\in\mathbb{N}$. Then
$\sup_{B^{n+1}_{\frac{R}{2}}(0)\cap\hat{M}_{j}}|A_{\hat{M}_{j}}|\to 0$ as
$j\to\infty$, and $\hat{M}_{j}$ converge smoothly to $P$ in
$B^{n+1}_{\frac{R}{2}}(0)$ (as immersions, with the limit being an immersion
with $q$ connected components, each covering $P$ once).

####  3.2 Closeness to a classical cone

Let $C$ be a classical cone, i.e. a sum of half-hyperplanes all intersecting
at a given $(n-1)$-dimensional subspace, $C=\sum_{i=1}^{N}q_{i}\llbracket
H_{i}\rrbracket$ with $q_{i}\in\mathbb{N}$, $H_{i}$ a half-hyperplane whose
boundary is the given $(n-1)$-dimensional subspace. Without loss of generality
we assume that the $(n-1)$-dimensional subspace is the span of
$\\{\textbf{e}_{3},\ldots,\textbf{e}_{n+1}\\}$. For $\tau>0$, let $C_{\tau}$
denote the cylinder $\\{x_{1}^{2}+x_{2}^{2}<\tau^{2}\\}$.

######  Lemma 3.2.

Let $\hat{M}_{j}$ be a sequence of smooth properly immersed two-sided stable
minimal hypersurfaces in $B_{4R}$. Assume that $\hat{M}_{j}$ converge (as
varifolds in $B_{4R}$) to $C$ as $j\to\infty$, where $C$ is a classical cone
as above. Then $\int_{B_{2R}\cap\hat{M}_{j}}|A_{\hat{M}_{j}}|^{2}\to 0$ as
$j\to\infty$.

###### Proof.

By scale-invariance we may take $4R=2$. By Lemma 3.1, for $\tau>0$ we have
that $\hat{M}_{j}$ converge strongly to $C$ in $B_{2}\setminus
C_{\frac{\tau}{2}}$. In particular, given $\tau>0$, we have
$\int_{\big{(}B_{1}\setminus
C_{\tau}\big{)}\cap\hat{M}_{j}}|A_{\hat{M}_{j}}|^{2}\to 0$ as $j\to\infty$.
Further,

$\displaystyle\int_{\hat{M}_{j}\cap C_{\tau}\cap
B_{1}}|A_{\hat{M}_{j}}|^{2}\leq\left(\int_{\hat{M}_{j}\cap C_{\tau}\cap
B_{1}}|A_{\hat{M}_{j}}|^{\frac{2n}{n-2}}\right)^{\frac{n-2}{n}}\left(\int_{\hat{M}_{j}\cap
C_{\tau}\cap B_{1}}1\right)^{\frac{2}{n}}$
$\displaystyle\leq\left(\int_{\hat{M}_{j}\cap
B_{1}}|A_{\hat{M}_{j}}|^{\frac{2n}{n-2}}\right)^{\frac{n-2}{n}}\mathcal{H}^{n}(\hat{M}_{j}\cap
C_{\tau}\cap B_{1})^{\frac{2}{n}}.$

Lemma 2.1 taken with $k=0$ implies (as argued for (11), for a dimensional
$K(n)$)

$\int_{\hat{M}_{j}\cap B_{1}}|A_{\hat{M}_{j}}|^{\frac{2n}{n-2}}\leq
K(n)\left(\int_{\hat{M}_{j}\cap
B_{\frac{3}{2}}}|A_{\hat{M}_{j}}|^{2}\right)^{\frac{n}{n-2}}.$

Letting $\eta$ be a fixed function that is equal to $1$ in $B_{\frac{3}{2}}$,
is supported in $B_{2}$, and with $|\nabla\eta|\leq 4$, we find
$\int_{\hat{M}_{j}\cap
B_{\frac{3}{2}}}|A_{\hat{M}_{j}}|^{2}\leq\int_{\hat{M}_{j}\cap
B_{2}}|\nabla\eta|^{2}\leq 16\Lambda(C)$, for all sufficiently large $\ell$
(we used the stability inequality), where we let $\Lambda(C)=\|C\|(B_{2})+1$.
Therefore,
$\left(\int_{B_{1}}|A_{\hat{M}_{j}}|^{\frac{2n}{n-2}}\right)^{\frac{n-2}{n}}$
is uniformly bounded for all sufficiently large $\ell$. Next, observe that
$C_{\tau}\cap B_{1}$ is contained in the union of $\frac{b(n)}{\tau^{n-1}}$
balls of radius $\tau$, where $b(n)$ is a dimensional constant (a rough cover
shows that $b(n)=n^{\frac{n-1}{n}}$ works), and in each such ball the $n$-area
of $\hat{M}_{j}$ is at most $k(n)\Lambda(C)\tau^{n}$, by the monotonicity
formula for the area ratio, for a dimensional constant $k(n)$. Hence
$\mathcal{H}^{n}(\hat{M}_{j}\cap C_{\tau}\cap
B_{1})^{\frac{2}{n}}\leq\Big{(}\Lambda(C)b(n)k(n)\Big{)}^{\frac{2}{n}}\tau^{\frac{2}{n}}$
and

$\int_{B_{1}\cap\hat{M}_{j}}|A_{\hat{M}_{j}}|^{2}\leq\Bigg{(}\int_{(B_{1}\setminus
C_{\tau})\cap\hat{M}_{j}}|A_{\hat{M}_{j}}|^{2}\Bigg{)}+\Lambda(C)b(n)k(n)\tau^{\frac{2}{n}}.$

As $\tau>0$ can be chosen arbitrarily small,
$\int_{B_{1}\cap\hat{M}_{j}}|A_{\hat{M}_{j}}|^{2}\to 0$ as $j\to\infty$. ∎

This shows that, for all sufficiently large $j$, the quantity
$\int_{B_{2R}^{n+1}\cap\hat{M}_{j}}|A_{\hat{M}_{j}}|^{2}$ is at most
$\epsilon_{0}$, so Theorem 3 applies. As immediate consequence, we have:

######  Lemma 3.3.

Let $\hat{M}_{j}$ be a sequence of smooth properly immersed two-sided stable
minimal hypersurfaces that converge (as varifolds) in $B^{n+1}_{4R}(0)$ to a
classical cone $C$, as $j\to\infty$, with $n\leq 6$. Then $C$ is a sum of
hyperplanes with multiplicity, which we describe as a smooth immersion;
moreover, $\sup_{B^{n+1}_{\frac{R}{2}}(0)\cap\hat{M}_{j}}|A_{\hat{M}_{j}}|\to
0$ as $j\to\infty$, and $\hat{M}_{j}$ converge smoothly to $C$ in
$B^{n+1}_{\frac{R}{2}}(0)$ as immersions (with $q$ connected components, with
$q=\Theta(\|C\|,0)$).

####  3.3 Tangent cone analysis and conclusion

######  Lemma 3.4.

Let $M_{j}$ be a sequence of smooth properly immersed two-sided stable minimal
hypersurfaces in an open set $U\subset\mathbb{R}^{n+1}$, with $n\leq 6$,
converging (as varifolds) to a (stationary integral) varifold $V$. Let $x\in
U$ be such that (at least) one tangent cone to $V$ at $x$ is either a
hyperplane with multiplicity, or a classical cone. There exists $\rho>0$ such
that $V$ is the varifold associated to a smooth immersion in
$B_{\rho}^{n+1}(x)$, and $M_{j}$ converge smoothly (as immersions) to $V$ in
$B_{\rho}^{n+1}(x)$.

###### Proof.

It suffices to find $\rho>0$ such that $\limsup_{j\to\infty}\sup_{M_{j}\cap
B^{n+1}_{2\rho}(x)}|A_{M_{j}}|<\infty$ and $2\rho<\text{dist}(x,\partial U)$
(after which, standard compactness under $L^{\infty}$ curvature bounds gives
the result). Arguing by contradiction, we assume that this fails for every
such $\rho$, hence there exist a subsequence (not relabeled) $M_{j}$ and
associated points $x_{j}\in M_{j}$ with $|A_{M_{j}}(x_{j})|\to\infty$ and
$x_{j}\to x$. Upon passing to a further subsequence (determined by the chosen
blow up of $V$ at $x$), we find, for each $j$, rescalings $\tilde{M}_{j}$ of
$M_{j}$ around $x$ that converge to the chosen tangent to $V$ at $x$ (either a
hyperplane with multiplicity, or a classical cone) and such that
$|A_{\tilde{M}_{j}}(\tilde{x}_{j})|\to\infty$, where
$\tilde{x}_{j}\in\tilde{M}_{j}$ is the image of $x_{j}$ via the dilation
associated to $j$. We apply Lemma 3.1 if the tangent is a hyperplane with
multiplicity, and Lemma 3.3 if the tangent is a classical cone, reaching a
contradiction in both cases. ∎

The argument that we recall next is a classical procedure (see e.g. [13,
Section 6], [18, Section 8]) that involves tangent cone analysis (in the sense
of varifolds), Federer’s dimension reducing principle (see e.g. [16, Appendix
A]), and Simons’ classification of stable cones ([17], see also [16, Appendix
B] and [14, Section 3]), together with Lemma 3.4 itself, to show that:

######  Proposition 3.1.

Let $U$, $M_{j}$, $V$ be as in the hypotheses of Lemma 3.4. Then the only
possible tangent cones to $V$ are hyperplanes with multiplicity and classical
cones. In particular, the conclusion of Lemma 3.4 applies at every
$x\in\text{spt}\|V\|$.

It turns out that it is natural to prove the stronger result that any iterated
tangent to $V$ is a hyperplane with multiplicity or classical cone. We first
recall the relevant notions and facts.

If $M_{\ell}\to V$ and $C_{x}$ is a tangent cone to $V$ at $x$, there exist
$r_{j}\to 0$ and a subsequence $\ell(j)$ such that
$\hat{M}_{\ell(j)}=\lambda_{\big{(}x,\frac{1}{r_{j}}\big{)}}M_{\ell(j)}$
converge (as varifolds) to $C_{x}$, where
$\lambda_{\big{(}x,\frac{1}{r_{j}}\big{)}}$ is the dilation of factor
$\frac{1}{r_{j}}$ centred at $x$, combined with the translation that sends $x$
to $0$, that is
$\lambda_{\big{(}x,\frac{1}{r_{j}}\big{)}}(z)=\frac{z-x}{r_{j}}$.

The spine $S(C)$ of a cone $C$ is the maximal subspace of translation
invariance (and coincides with the set of points of maximal density). We
further recall the notion of iterated tangents (to $V$ at $x$), by which we
mean the collection of cones $C$ for which there exist cones
$C_{1},\ldots,C_{N}$, with $C_{N}=C$ and $N\in\mathbb{N}$, $N\geq 1$, and
points $p_{1}\in C_{1}\setminus S(C_{1}),\ldots p_{N}\in C_{N}\setminus
S(C_{N})$ such that $C_{m}$ is a tangent cone to $C_{m-1}$ at $p_{m-1}$ for
$m\geq 2$ and $C_{1}=C_{x}$ is a tangent cone to $V$ at $x$. For every $C$ in
the space of iterated tangents to $V$ at $x$, we can find $r_{j}\to 0$ a
subsequence $\ell(j)$ and points $z_{\ell(j)}\to x$ (not necessarily lying on
$M_{\ell(j)}$) such that
$\tilde{M}_{\ell(j)}=\lambda_{\big{(}z_{\ell(j)},\frac{1}{r_{j}}\big{)}}M_{\ell(j)}$
converge (as varifolds) to $C$, where
$\lambda_{\big{(}z_{\ell(j)},\frac{1}{r_{j}}\big{)}}$ is the dilation of
factor $\frac{1}{r_{j}}$ centred at $z_{\ell(j)}$, combined with the
translation that sends $z_{\ell(j)}$ to $0$, that is
$\lambda_{\big{(}z_{\ell(j)},\frac{1}{r_{j}}\big{)}}(z)=\frac{z-z_{\ell(j)}}{r_{j}}$.
(In the case $N=1$ one can take $z_{\ell(j)}=x$, as seen above.)

###### Proof of Proposition 3.1.

We first show that _for any iterated tangent $C$ the smoothly immersed part of
$C$ is stable_. Indeed, in a sufficiently small ambient ball
$B^{n+1}_{\rho_{y}}(y)$, centred at any given $y\in C$ around which $C$ is
smoothly immersed, the (dilated) hypersurfaces $\tilde{M}_{\ell(j)}$ converge
smoothly (as immersions) to $C$. This is a consequence of Lemma 3.4, as the
(unique, in this case) tangent to $C$ at $y$ is either a hyperplane with
multiplicity, or a classical cone. The arbitrariness of $y$ leads to smooth
convergence of $\tilde{M}_{\ell(j)}$ to $C$ on the smoothly immersed part of
$C$, and thus the stability condition is inherited by the smoothly immersed
part of $C$.

By Simons’ classification, there exists no cone $C$ of dimension
$n\in\\{2,\ldots,6\\}$, smoothly immersed away from an isolated singularity at
the tip, which is stable on the smoothly immersed part. This implies that
there cannot exist a cone $C$, in the above collection of iterated tangents,
that has spine of dimension $\leq n-2$ and is smoothly immersed away from the
spine — in other words, _the only (iterated tangent) cones that are smoothly
immersed away from the spine are hyperplanes with multiplicity and classical
cones_ (corresponding respectively to the cases of spine dimension $n$ and
$n-1$). To see this, assume that $C\setminus S(C)$ is smoothly immersed and
stable, with $\text{dim}S(C)\leq n-2$. Slice $C$ with affine planes of
dimension complementary to the spine $S(C)$, and orthogonal to it. Any such
slice is a cone, of dimension at least $2$, since $\text{dim}S(C)\leq(n-2)$,
and at most $n$, since it is a slice of the $n$-dimensional cone $C$. This
slice is a smoothly immersed cone except for an isolated singularity at the
tip; moreover, its regular part inherits stability. Since $n\leq 6$, this is
not possible by Simons’ result.

A key fact, underlying Federer’s dimension reducing principle, is that the
spine dimension strictly increases when we take iterated tangents,
$\text{dim}S(C_{1})<\ldots<\text{dim}S(C_{N})$ with the above notation. (This
is due to the fact that, choosing a point $y$ away from the spine $S(C)$, the
linear subspace spanned by $y$ and $S(C)$ becomes translation invariant for
the tangent to $C$ at $y$.) This is used in the following way, to prove that
_any iterated tangent must be smoothly immersed away from its spine_.

Assume that a given cone $C$ in the collection of iterated tangents is not
smoothly immersed away from its spine $S(C)$, whose dimension we denote by
$s$. As $C$ is neither a hyperplane with multiplicity nor a classical cone, we
must have $s\in\\{0,\ldots,n-2\\}$. We consider a tangent cone to $C$ at a
non-immersed point in $C\setminus S(C)$, and iterate this step until we find a
cone $\hat{C}$ that is smoothly immersed away from its spine. This is achieved
after at most $n-s-1$ iterations (thanks to the strict increase in spine
dimension, after $n-s-1$ iterations we must have a classical cone or a
hyperplane with multiplicity). We let $\tilde{C}$ be the iterated tangent cone
for which $\hat{C}$ is a tangent cone at a non-immersed point
$y\in\tilde{C}\setminus S(\tilde{C})$.

As shown above, $\hat{C}$ is either a hyperplane with multiplicity or a
classical cone. Lemma 3.4 applies to the sequence $\tilde{M}_{\ell(j)}$ that
converges to $\tilde{C}$ in a suitably small ball, contradicting that $y$ is a
non-immersed point. This concludes the proof of Proposition 3.1. ∎

Theorem 2 follows by a contradiction argument. Using standard compactness
arguments (which require the given mass bounds), we assume the existence of a
sequence $M_{\ell}$ of hypersurfaces in $B_{4}^{n+1}(0)$ (that satisfy the
same assumptions as $M$ in the theorem) and, arguing by contradiction, assume
that there exists $x_{\ell}\in M_{\ell}\cap B_{\frac{1}{2}}$ with
$\limsup_{\ell\to\infty}|A_{M_{\ell}}(x_{\ell})|=\infty$. Allard’s compactness
gives a (subsequential) stationary limit $V$ for $M_{\ell}$ (without
relabelling the subsequence), in the sense of varifolds. By extracting a
further subsequence (without relabelling) we also assume $x_{\ell}\to
x\in\text{spt}\|V\|$. Proposition 3.1 (and Lemma 3.4) applied at $x$
contradicts $\limsup_{\ell\to\infty}|A_{M_{\ell}}(x_{\ell})|=\infty$.

Theorem 1 follows by considering $M\cap B_{4R}^{n+1}(p)$ for any chosen $p\in
M$, and translating (sending $p$ to $0$). As $R\to\infty$, the estimate in
Theorem 2 remains valid with the same $\beta$. This forces $A_{M}(p)=0$. Hence
$A\equiv 0$ on $M$ and the result follows.

##  Part II Towards a compactness theory for branched stable minimal
immersions

We are interested, in this second part, in a wider class immersed
hypersurfaces $M$: we allow a singular set $\text{Sing}_{M}$ with locally
finite $\mathcal{H}^{n-2}$-measure, or, more generally, vanishing
$2$-capacity. Explicitly, for $U\subset\mathbb{R}^{n+1}$ open, and
$\Sigma\subset U$ closed (in $U$) with $\text{cap}_{2}(\Sigma)=0$ (in
particular333We refer to [7] for details on capacity. In our context, the
implication
$\mathcal{H}^{n-2}(\Sigma)<\infty\Rightarrow\text{cap}_{2}(\Sigma)=0$ is
implicitly proved in [13] when $\mathcal{H}^{n-2}(\Sigma)=0$ and refined in
[18] for the case $\mathcal{H}^{n-2}(\Sigma)<\infty$ using a Federer-Ziemer
argument., we allow $\Sigma$ to have locally finite
$\mathcal{H}^{n-2}$-measure, that is, $\mathcal{H}^{n-2}(\Sigma\cap K)<\infty$
for every $K\subset\subset U$), we let $\iota:S\to U\setminus\Sigma$ be a
(smooth) proper immersion, that we assume to be two-sided minimal and stable,
with continuous unit normal $\nu$. Denoting by $\overline{M}$ the closure of
$M$ in $U$, we say that $x\in\text{Sing}_{M}$ if, for every $r>0$,
$B_{r}^{n+1}(x)\cap\overline{M}$ is not the image of a smooth immersion. (In
other words, a point in $\Sigma$ is genuinely singular if $M$ cannot be
smoothly extended across it, as an immersion.)

As proved in [13, (1.18) and Section 5], the stationarity condition (with
respect to the area functional) is valid for ambient deformations in $U$, that
is, the integral varifold $|\iota_{\sharp}S|$ is stationary in $U$ (not only
in $U\setminus\Sigma$). This follows from a suitable extension of the
monotonicity formula, obtained at points in $\Sigma$, giving Euclidean area
growth around all points in $\overline{M}\cap U$, combined with a standard
capacity argument. (In fact, [13] shows that $\mathcal{H}^{n-1}(\Sigma)=0$
would be sufficient for this.)

###  4 Proof of Theorem 4

_The tilt function and the relevant PDE_. For a given fixed unit vector, that
we assume without loss of generality to be the last coordinate vector
$e_{n+1}$, consider the function $g=\left(1-(\nu\cdot
e_{n+1})^{2}\right)^{1/2}$, well-defined on $S$. Clearly, $0\leq g\leq 1$.
Letting $\nabla$ denote the metric gradient on $S$, it is immediate that
$|\nabla g|\leq\sqrt{1-g^{2}}|A|$. This follows by direct computation, since

$|\nabla(\nu\cdot e_{n+1})|=|(D\nu)(e_{n+1}^{T})|\leq|A||e_{n+1}-(\nu\cdot
e_{n+1})\nu|=|A|g,$

where $e_{n+1}^{T}$ denotes the tangential part of $e_{n+1}$ and $D\nu$ is the
shape operator, and

$\nabla g=\frac{-(\nu\cdot e_{n+1})\nabla(\nu\cdot e_{n+1})}{\sqrt{1-(\nu\cdot
e_{n+1})^{2}}},\text{ that is, }g^{2}|\nabla g|^{2}=(1-g^{2})|\nabla(\nu\cdot
e_{n+1})|^{2}.$ (13)

We recall the standard Jacobi field equation $\Delta(\nu\cdot
e_{n+1})=-|A|^{2}(\nu\cdot e_{n+1})$, or, equivalently,

$-\Delta(\nu\cdot e_{n+1})^{2}=-2|\nabla(\nu\cdot
e_{n+1})|^{2}+2|A|^{2}(\nu\cdot e_{n+1})^{2},$

where $\Delta$ is the Laplace-Beltrami operator on $S$. This implies a PDE for
$g$ on $S$ (using the relation (13)):

$(1-g^{2})(2g\Delta g+2|\nabla g|^{2})=(1-g^{2})\big{(}\Delta
g^{2}\big{)}=-2g^{2}|\nabla g|^{2}+2|A|^{2}(1-g^{2})^{2},$

and therefore (in view of $|\nabla g|^{2}\leq(1-g^{2})|A|^{2}$ the following
is well-defined)

$g\Delta g=-\frac{|\nabla g|^{2}}{1-g^{2}}+|A|^{2}(1-g^{2}).$ (14)

We recall that the following improved inequality (see [13, (2.7)]) is implied
by the minimality condition:

$\frac{|\nabla(\nu\cdot e_{n+1})|^{2}}{(1-(\nu\cdot
e_{n+1})^{2})}\leq\left(1-\frac{1}{n}\right)|A|^{2}\Leftrightarrow\frac{|\nabla
g|^{2}}{1-g^{2}}\leq\left(1-\frac{1}{n}\right)|A|^{2}.$ (15)

######  Remark 4.1.

The quantity $E_{M}(R)^{2}=\frac{1}{R^{n}}\int_{C_{R}}g^{2}$ (appearing in
Theorem 4) is the square of the scale-invariant tilt-excess of $|M|$ in
$C_{R}=B^{n}_{R}(0)\times(-R,R)$, with respect to the hyperplane
$\mathbb{R}^{n}\times\\{0\\}$ (orthogonal to $e_{n+1}$). As in Part I, with
slight notational abuse we will write domains $D$, or $M\cap D$, to mean
$\iota^{-1}(D)$, where $\iota:S\to C_{2R}$ is the immersion with image $M$.

######  Remark 4.2 (_height and tilt excess_).

We recall that the (scale-invariant) $L^{2}$ height-excess $\hat{E}_{M}(r)$,
defined by $\hat{E}_{M}(r)^{2}=\frac{1}{r^{n+2}}\int_{M\cap
C_{r}}|x_{n+1}|^{2}$, bounds (linearly) $E_{M}(\frac{r}{2})^{2}$. Indeed,
stationarity implies, using the first variation formula with a vector field
$x_{n+1}\varphi^{2}e_{n+1}$, for $\varphi\in C^{1}_{c}(C_{2R})$ taken to be
identically $1$ in $C_{R}$ and with $|\nabla\varphi|\leq\frac{1}{R}$, the
validity of the inequality (see e.g. [16, Section 22])

$\frac{1}{R^{n}}\int_{M\cap C_{R}}|\nabla
x_{n+1}|^{2}\leq\frac{2^{n+4}}{(2R)^{n+2}}\int_{M\cap C_{2R}}|x_{n+1}|^{2},$

and $|\nabla x_{n+1}|=|\text{proj}_{TM}(e_{n+1})|=\sqrt{1-(\nu\cdot
e_{n+1})^{2}}=g$.

The proof of Theorem 4 will be carried out by means of an iteration à la De
Giorgi, for which the fundamental lemma is an intrinsic weak Caccioppoli
inequality, for level set truncations of $g$ (Lemmas 4.1 and 4.2 below).

######  Lemma 4.1.

Let $M$ be as above. Then for any $k\in[0,\frac{1}{2n}]$ and $\phi\in
C^{0,1}_{c}(S)$ we have

$\frac{1}{2n}\int_{\\{g>k\\}}|\nabla
g|^{2}\left(1-\frac{k}{g}\right)\phi^{2}\leq\int{(g-k)^{+}}^{2}|\nabla\phi|^{2},$

where $(g-k)^{+}$ denotes the function
$(g-k)^{+}=\left\\{\begin{array}[]{ccc}g-k&\text{ when }g>k\\\ 0&\text{ when
}g\leq k\end{array}\right.$.

###### Proof.

We use the stability condition, whose analytic form is the validity of

$\int|A|^{2}\eta^{2}\leq\int|\nabla\eta|^{2}$

for all $\eta\in C^{1}_{c}(S)$. A standard approximation argument implies that
$\eta\in C^{0,1}_{c}(S)$ is also allowed and we choose $\eta=(g-k)^{+}\phi$,
where $\phi\in C^{0,1}_{c}(S)$ (as in the statement). We compute (on the
right-hand-side of the stability inequality)

$\displaystyle\int|\nabla((g-k)^{+}\phi)|^{2}=$
$\displaystyle\int|\nabla(g-k)^{+}|^{2}\phi^{2}+\underbrace{2\int(g-k)^{+}\,\phi\,\nabla(g-k)^{+}\,\nabla\phi}_{\frac{1}{2}\int\nabla({(g-k)^{+}}^{2})\,\nabla(\phi^{2})}+\int{(g-k)^{+}}^{2}|\nabla\phi|^{2}.$

We note that the function ${(g-k)^{+}}^{2}$ is in $C^{1}\cap W^{2,\infty}(S)$.
Indeed, $\nabla({(g-k)^{+}}^{2})=2{(g-k)^{+}}\nabla g$ and this function is
locally Lipschitz. In particular, we have that $\Delta({(g-k)^{+}}^{2})$ is
the $L^{\infty}$ function that vanishes in the complement of $\\{g\geq k\\}$
and is equal to $2{(g-k)^{+}}\Delta{(g-k)^{+}}+2|\nabla({(g-k)^{+}}|^{2}$ on
$\\{g<k\\}$. Hence we can integrate by parts and the braced term becomes

$-\frac{1}{2}\int\Delta({(g-k)^{+}}^{2})\,\phi^{2}=-\int|\nabla{(g-k)^{+}}|^{2}\,\phi^{2}-\int_{\\{g>k\\}}{(g-k)^{+}}\Delta{(g-k)^{+}}\,\phi^{2}.$

The right-hand-side of the stability inequality is therefore

$\displaystyle-\int_{\\{g>k\\}}(g-k)\Delta
g\,\phi^{2}+\int{(g-k)^{+}}^{2}|\nabla\phi|^{2}\stackrel{{\scriptstyle\text{
by }\eqref{eq:PDE_for_g}}}{{=}}$
$\displaystyle\int_{\\{g>k\\}}\Bigg{(}1-\frac{k}{g}\Bigg{)}\frac{|\nabla
g|^{2}}{1-g^{2}}\,\phi^{2}-\int_{\\{g>k\\}}\Bigg{(}1-\frac{k}{g}\Bigg{)}|A|^{2}(1-g^{2})\,\phi^{2}+\int{(g-k)^{+}}^{2}|\nabla\phi|^{2}.$

(When $k=0$ we do not need to multiply the PDE (14) by
$\frac{g-k}{g}=1-\frac{k}{g}$.) We now use the improved inequality (15) (for
the first integrand in the last expression) and find, from the stability
inequality,

$\displaystyle\int_{\\{g>k\\}}|A|^{2}(g-k)^{2}\phi^{2}\leq\int_{\\{g>k\\}}\Bigg{(}1-\frac{1}{n}\Bigg{)}\Bigg{(}1-\frac{k}{g}\Bigg{)}|A|^{2}\,\phi^{2}$
$\displaystyle-\int_{\\{g>k\\}}\Bigg{(}1-\frac{k}{g}\Bigg{)}|A|^{2}(1-g^{2})\,\phi^{2}+\int{(g-k)^{+}}^{2}|\nabla\phi|^{2}.$

Moving all terms containing $|A|^{2}$ to the left-hand-side we compute

$\displaystyle(g-k)^{2}-\Bigg{(}1-\frac{1}{n}\Bigg{)}\Bigg{(}1-\frac{k}{g}\Bigg{)}+\Bigg{(}1-\frac{k}{g}\Bigg{)}(1-g^{2})=$
$\displaystyle(g-k)\Bigg{(}g-k-\frac{1}{g}\Big{(}1-\frac{1}{n}-1+g^{2}\Big{)}\Bigg{)}=\frac{g-k}{g}\bigg{(}g^{2}-kg+\frac{1}{n}-g^{2}\bigg{)},$

which gives

$\int_{\\{g>k\\}}|A|^{2}\,\frac{(g-k)}{g}\left(\frac{1}{n}-kg\right)\,\phi^{2}\leq\int{(g-k)^{+}}^{2}|\nabla\phi|^{2}.$

As $g\in[0,1]$, if we restrict $k\in[0,\frac{1}{2n}]$ as in the hypotheses we
get $\frac{1}{n}-kg\geq\frac{1}{2n}$, hence

$\frac{1}{2n}\int_{\\{g>k\\}}|A|^{2}\left(1-\frac{k}{g}\right)\phi^{2}\leq\int{(g-k)^{+}}^{2}|\nabla\phi|^{2}.$
(16)

Using $|\nabla g|\leq|A|$ we reach

$\frac{1}{2n}\int_{\\{g>k\\}}|\nabla
g|^{2}\left(1-\frac{k}{g}\right)\phi^{2}\leq\int{(g-k)^{+}}^{2}|\nabla\phi|^{2}.$

∎

######  Lemma 4.2.

Let $M$ be as above. Then for any $k\in[0,\frac{1}{2n}]$ and $\varphi\in
C^{0,1}_{c}(U)$ we have

$\frac{1}{2n}\int_{M\cap\\{g>k\\}}|\nabla
g|^{2}\left(1-\frac{k}{g}\right)\varphi^{2}\leq\int_{M}{(g-k)^{+}}^{2}|\nabla\varphi|^{2},$

where $(g-k)^{+}$ denotes the function
$(g-k)^{+}=\left\\{\begin{array}[]{ccc}g-k&\text{ when }g>k\\\ 0&\text{ when
}g\leq k\end{array}\right.$.

###### Proof.

The statement is just Lemma 4.1 when $\varphi\in
C^{0,1}_{c}(U\setminus\Sigma)$. (We are implicitly choosing
$\phi=\varphi\circ\iota$; the immersion is proper so $\varphi\circ\iota\in
C^{1}_{c}(S)$.) Taking this as starting point, the extension of the inequality
to $\varphi\in C^{0,1}_{c}(U)$ relies on the Euclidean area growth of $n$-area
(valid at all points in $\overline{M}$, as recalled above) and on the
assumption that $\text{cap}_{2}(\Sigma)=0$. The (now standard) $2$-capacity
argument is carried out in [13] for the case $\mathcal{H}^{n-2}(\Sigma)=0$ and
in [18] for the case $\mathcal{H}^{n-2}(\Sigma)<\infty$. ∎

######  Remark 4.3.

In the case $k=0$, (16) is the well-known Schoen inequality, [12], [13, Lemma
1]. The instance $k=0$ of the lemma gives the intrinsic Caccioppoli inequality
$\frac{1}{2n}\int|\nabla g|^{2}\varphi^{2}\leq\int g^{2}|\nabla\varphi|^{2}$.

We will employ Lemma 4.2, with suitable choices of $\varphi$. We will obtain a
superlinear rate of decay for the $L^{2}$-norm of $(g-k)^{+}$ in $C_{r}$ as
$k\in\mathbb{R}$ grows from $0$ to $\frac{1}{2n}$, and $r$ decreases from the
initial scale $R$ to $\frac{R}{2}$. Via the elementary Lemma B.1, such a decay
forces $(g-\frac{1}{2n})^{+}$ to vanish in $C_{\frac{R}{2}}$, as long as the
$L^{2}$-norm of $g$ is sufficiently small in $C_{R}$. This will establish
Theorem 4.

###### proof of Theorem 4 for $n\geq 3$.

We consider the dyadic sequences (respectively increasing and decreasing)
$k_{\ell}=\frac{d}{2n}\left(1-\frac{1}{2^{\ell-1}}\right)$ for $d\in(0,1]$
(for the moment arbitrary), and $R_{\ell}=\frac{R}{2}+\frac{R}{2^{\ell}}$ for
$\ell\in\\{1,2,\ldots\\}$.

We take the inequality of Lemma 4.2 with $k_{\ell}$ in place of $k$, and use
the inclusion $\\{g>k_{\ell}\\}\supset\\{g>k_{\ell+1}\\}$:

$\frac{1}{2n}\int_{\\{g>k_{\ell+1}\\}}|\nabla
g|^{2}\left(1-\frac{k_{\ell}}{g}\right)\varphi^{2}\leq\int{(g-k_{\ell})^{+}}^{2}|\nabla\varphi|^{2};$

on the relevant domain on the left-hand-side, $\\{g>k_{\ell+1}\\}$, we have
$1-\frac{k_{\ell}}{g}=\frac{g-k_{\ell}}{g}>\frac{k_{\ell+1}-k_{\ell}}{g}=\frac{d}{2^{\ell}2ng}\geq\frac{d}{2^{\ell+1}n}$,
therefore

$\int_{\\{g>k_{\ell+1}\\}}|\nabla
g|^{2}\varphi^{2}\leq\frac{4n^{2}2^{\ell}}{d}\int{(g-k_{\ell})^{+}}^{2}|\nabla\varphi|^{2}.$

Since $|\nabla\big{(}(g-k_{\ell+1})^{+}\varphi\big{)}|^{2}\leq
2\chi_{\\{g>k_{\ell+1}\\}}|\nabla
g|^{2}\varphi^{2}+2{(g-k_{\ell+1})^{+}}^{2}|\nabla\varphi|^{2}$, and by
definition $(g-k_{\ell+1})^{+}\leq(g-k_{\ell})^{+}$, we find

$\int_{\\{g>k_{\ell+1}\\}}\Big{|}\nabla\big{(}(g-k_{\ell+1})^{+}\varphi\big{)}\Big{|}^{2}\leq\frac{2(4n^{2}2^{\ell}+1)}{d}\int{(g-k_{\ell})^{+}}^{2}|\nabla\varphi|^{2}.$
(17)

Next (from now we will use $n\geq 3$), we will use (for the left-hand-side of
(17)) the following Michael-Simon inequality on the minimally immersed
hypersurface $M$, for the function $\varphi(g-k_{\ell+1})^{+}$:

$\left(\int\left|\varphi(g-k_{\ell+1})^{+}\right|^{\frac{2n}{n-2}}\right)^{\frac{n-2}{n}}\leq
C_{MS}\int\left|\nabla\left(\varphi(g-k_{\ell+1})^{+}\right)\right|^{2},$ (18)

with $C_{MS}$ the dimensional constant given after (7).

Simultaneously, we choose $\varphi$, as follows. For $r>\rho$ chosen in
$(0,R]$ we will consider $\varphi$ of the type
$\varphi(x,x_{n+1})=\tilde{\varphi}(x)\tilde{\psi}(x_{n+1})$, with
$\tilde{\varphi}:\mathbb{R}^{n}\to\mathbb{R}$ identically equal to $1$ on
$B_{\rho}^{n}(0)$, vanishing in the complement of $B_{r}^{n}(0)$ and with
$|D\tilde{\varphi}|\leq\frac{\sqrt{2}}{r-\rho}$; with $\tilde{\psi}\in
C^{\infty}_{c}(\mathbb{R})$ identically equal to $1$ on $[-\rho,\rho]$,
vanishing in the complement of $(-r,r)$, with
$|\tilde{\psi}^{\prime}|\leq\frac{\sqrt{2}}{r-\rho}$. Then, for each $\ell$,
we choose $\tilde{\varphi}_{\ell}$ and $\tilde{\psi}_{\ell}$ with
$\rho=R_{\ell+1}$, $r=R_{\ell}$, so that
$r-\rho=R_{\ell}-R_{\ell+1}=\frac{R}{2^{\ell+1}}$, and
$\varphi_{\ell}=\tilde{\varphi}_{\ell}\tilde{\psi}_{\ell}$. Note that
$|\nabla\varphi_{\ell}|\leq\frac{2}{R_{\ell}-R_{\ell+1}}$,
$\varphi_{\ell}\equiv 1$ on $C_{R_{\ell+1}}$ and $\varphi_{\ell}\equiv 0$ in
the complement of $C_{R_{\ell}}$. Combining (17) and (18), with the chosen
$\varphi_{\ell}$ in place of $\varphi$, we find

$\displaystyle\left(\int_{M\cap
C_{R_{\ell+1}}}\left|(g-k_{\ell+1})^{+}\right|^{\frac{2n}{n-2}}\right)^{\frac{n-2}{n}}\leq\left(\int\left|\varphi_{\ell}(g-k_{\ell+1})^{+}\right|^{\frac{2n}{n-2}}\right)^{\frac{n-2}{n}}\leq$
(19) $\displaystyle\leq
C_{MS}(n)\frac{2(4n^{2}2^{\ell}+1)4^{\ell+2}}{dR^{2}}\int_{M\cap
C_{R_{\ell}}}{(g-k_{\ell})^{+}}^{2}.$

Hölder’s inequality further gives

$\displaystyle\int_{M\cap C_{R_{\ell+1}}}{(g-k_{\ell+1})^{+}}^{2}\leq$ (20)
$\displaystyle\left(\int_{M\cap
C_{R_{\ell+1}}}{(g-k_{\ell+1})^{+}}^{\frac{2n}{n-2}}\right)^{\frac{n-2}{n}}\mathcal{H}^{n}\Big{(}\\{g>k_{\ell+1}\\}\cap
C_{R_{\ell+1}}\Big{)}^{\frac{2}{n}}.$

Noting that on the set $\\{g>k_{\ell+1}\\}$ we have
$(g-k_{\ell})^{+}>\frac{d}{n\,2^{\ell+1}}$, and using the inclusion
$C_{R_{\ell+1}}\subset C_{R_{\ell}}$, the last factor in (20) is bounded above
(thanks to the standard Markov’s inequality) by

$\displaystyle\mathcal{H}^{n}\Bigg{(}M\cap\left\\{{(g-k_{\ell})^{+}}^{2}>\frac{d^{2}}{(n\,2^{\ell+1})^{2}}\right\\}\cap
C_{R_{\ell}}\Bigg{)}^{2/n}\leq$ (21)
$\displaystyle\left(\frac{n^{2}\,4^{\ell+1}}{d^{2}}\int_{M\cap
C_{R_{\ell}}}{(g-k_{\ell})^{+}}^{2}\right)^{2/n}.$

From (20), using (21) for the second factor on the right-hand-side, and using
(19) for the first factor on the right-hand-side, we have

$\displaystyle\int_{M\cap C_{R_{\ell+1}}}{(g-k_{\ell+1})^{+}}^{2}\leq$
$\displaystyle
C_{MS}(n)\frac{8(4n^{2}2^{\ell}+1)4^{\ell+1}}{d^{1+\frac{4}{n}}R^{2}}n^{\frac{4}{n}}(4^{\frac{2}{n}})^{\ell+1}\Big{(}\int_{M\cap
C_{R_{\ell}}}{(g-k_{\ell})^{+}}^{2}\Big{)}^{1+\frac{2}{n}}.$

Writing $G_{\ell}=\int_{C_{R_{\ell}}}{(g-k_{\ell})^{+}}^{2}$, this implies

$G_{\ell+1}\leq\underbrace{C_{MS}(n)\frac{64n^{2}}{d^{1+\frac{4}{n}}R^{2}}n^{4/n}(4\cdot
4^{2/n})}_{c(n,R,d)}\,(8\cdot 4^{2/n})^{\ell}\,G_{\ell}^{1+\frac{2}{n}}.$ (22)

The superlinear decay estimate (22), $G_{\ell+1}\leq
c(n,R,d)C^{\ell}G_{\ell}^{1+\frac{2}{n}}$ with $C=2\cdot 4^{1+\frac{2}{n}}$,
forces $G_{\ell}\to 0$ as $\ell\to\infty$, as long as $G_{1}$ is sufficiently
small, in a quantified fashion determined by $c(n,R,d)$ and $C$. We make it
now explicit, using Lemma B.1.

With the initial choice $d=1$, the smallness condition on $G_{1}$ is written,
for the scale-invariant tilt-excess ($R_{1}=R$ so
$E_{M}(R)^{2}=\frac{1}{R^{n}}G_{1}$), as

$E_{M}(R)^{2}\leq\frac{1}{\left(R^{2}c(n,R,1)C^{\frac{n+2}{2}}\right)^{n/2}}=\left(\frac{1}{C_{MS}(n)64\,n^{2+\frac{4}{n}}4^{1+\frac{2}{n}}2^{\frac{n+2}{2}}4^{\frac{(n+2)^{2}}{2n}}}\right)^{n/2},$
(23)

where the last term makes explicit the dimensional constant $k(n)$ in Theorem
4.

The convergence $G_{\ell}\to 0$ implies
$\int_{C_{\frac{R}{2}}}{(g-\frac{1}{2n})^{+}}^{2}=0$, that is,
$g\leq\frac{1}{2n}$ a.e. on $M\cap C_{\frac{R}{2}}$. By smoothness of $\iota$,
then $g\leq\frac{1}{2n}$ on $M\cap C_{\frac{R}{2}}$.

More generally, with $d\in(0,1]$, we find that, if

$E_{M}(R)^{2}\leq\frac{1}{\left(R^{2}c(n,R,d)C^{\frac{n+2}{2}}\right)^{n/2}}=\frac{d^{2+\frac{n}{2}}}{\Big{(}C_{MS}(n)64\,n^{2+\frac{4}{n}}4^{1+\frac{2}{n}}2^{\frac{n+2}{2}}4^{\frac{(n+2)^{2}}{2n}}\Big{)}^{n/2}},$

then $g\leq\frac{d}{2n}$ on $M\cap C_{\frac{R}{2}}$. In other words, we have
proved that, in the regime $E_{M}(R)^{2}\leq k(n)$, we have (for an explicit
dimensional constant $c(n)$)

$\sup_{M\cap C_{\frac{R}{2}}}g\leq c(n)E_{M}(R)^{\frac{4}{4+n}}.$

∎

######  Remark 4.4.

For $k=0$, the inequality in Lemma 4.2 is an _intrinsic_ Caccioppoli
inequality (we have the intrinsic gradient on $M$, rather than the gradient
$D$ in $\mathbb{R}^{n}$ as in the case of De Giorgi [5], see also Remark 2.4).
For $k\in(0,\frac{1}{2n}]$, on the other hand, we only have a weak intrinsic
Caccioppoli inequality, due to the multiplicative factor
$\left(1-\frac{k}{g}\right)$. As seen also for Lemma 2.1, this weaker
inequality is sufficient to implement the iterative scheme. While in [5] it is
the linearity of the PDE that permits to obtain the classical Caccioppoli
inequality for $(u-k)^{+}$, in our case the PDE for $g$ escapes the De
Giorgi–Nash–Moser framework: in fact, the PDE is a consequence of the
minimality of $M$ alone, which would permit e.g. catenoidal necks, with $g$
reaching the value $1$ under any smallness assumption on the $L^{2}$ height-
or tilt-excess. The stability condition provides sufficient control on the
non-linearity of the PDE (14) to obtain the weak intrinsic Caccioppoli
inequality. We note explicitly that Lemma 4.2 is only valid for truncations at
sufficiently small level sets (hence the smallness requirement in Theorem 4).

###### proof of Theorem 4 for $n=2$.

The case $n=2$ requires a modification, as the exponent $\frac{2n}{n-2}$ is
not well-defined in that case. The choices of $k_{\ell}$, $R_{\ell}$,
$\varphi_{\ell}$ remain the same. We start from (17) (only after which we used
$n\geq 3$), choosing $\varphi$ in (17) to be $\varphi_{\ell}$ (recall that
$|\nabla\varphi_{\ell}|\leq\frac{2^{\ell+1}}{R}$ and that
$\text{spt}\varphi_{\ell}\subset C_{R_{\ell}}$). In what follows,
$\sigma,\sigma^{\prime},\sigma^{\prime\prime}$ denote explicitly determinable
constants. We have

$\int\Big{|}\nabla\big{(}(g-k_{\ell+1})^{+}\varphi_{\ell}\big{)}\Big{|}^{2}\leq\frac{4^{\ell}\sigma}{dR^{2}}\int_{C_{R_{\ell}}\cap
M}{(g-k_{\ell})^{+}}^{2}.$

We use Hölder’s inequality

$\Big{(}\int\Big{|}\nabla\big{(}(g-k_{\ell+1})^{+}\varphi_{\ell}\big{)}\Big{|}\Big{)}^{2}\leq\Big{(}\int\Big{|}\nabla\big{(}(g-k_{\ell+1})^{+}\varphi_{\ell}\big{)}\Big{|}^{2}\Big{)}\mathcal{H}^{2}\big{(}M\cap\\{g>k_{\ell+1}\\}\cap
C_{R_{\ell}}\big{)},$

the Michael–Simon inequality

$\int\big{(}(g-k_{\ell+1})^{+}\varphi\big{)}^{2}\leq
C_{MS}^{2}\Big{(}\int\Big{|}\nabla\big{(}(g-k_{\ell+1})^{+}\varphi\big{)}\Big{|}\Big{)}^{2}$

and the following consequence of Markov’s inequality (as justified earlier)

$\mathcal{H}^{2}\big{(}M\cap\\{g>k_{\ell+1}\\}\cap
C_{R_{\ell}}\big{)}\leq\frac{\sigma^{\prime}4^{\ell}}{d^{2}}\int_{M\cap
C_{R_{\ell}}}{(g-k_{\ell})^{+}}^{2}.$

Writing $G_{\ell}=\int_{M\cap C_{R_{\ell}}}{(g-k_{\ell})^{+}}^{2}$, combining
these inequalities we find

$G_{\ell+1}\leq\frac{16^{\ell}\sigma^{\prime\prime}}{R^{2}d^{3}}G_{\ell}^{2}.$

At this stage, Lemma B.1 gives that, if
$G_{1}\leq\frac{R^{2}d^{3}}{16\sigma^{\prime\prime}}$ then $G_{\ell}\to 0$ as
$\ell\to\infty$. In other words, given $d\in(0,1]$, if
$E_{M}^{2}(R)=\frac{1}{R^{2}}\int_{M\cap
C_{R}}g^{2}\leq\frac{d^{3}}{16\sigma^{\prime\prime}}$, then $\sup_{M\cap
C_{\frac{R}{2}}}g\leq\frac{d}{2n}$. The conclusion of Theorem 4 is thus proved
for $n=2$: in the smallness regime
$E_{M}^{2}(R)\leq\frac{1}{16\sigma^{\prime\prime}}$ we have the control
$\sup_{M\cap C_{\frac{R}{2}}}g\leq c(n)E_{M}(R)^{\frac{2}{3}}$. ∎

###  5 Proof of Theorems 5, 6, 7

###### Proof of Theorem 6.

The pointwise bound $g\leq\frac{1}{2n}$ obtained in Theorem 4 implies the
decomposition result by elementary arguments. Being an immersion, $\iota$ is
locally a diffeomorphism with its image, that is, for every $X\in S$ there
exists a neighbourhood $D_{X}$ such that $\left.\iota\right|_{D_{X}}$ is an
embedded disk. The bound on $g$ implies that that there exists a choice of
continuous unit normal $\nu$ such that $\nu\cdot
e_{n+1}\geq\frac{\sqrt{(2n)^{2}-1}}{2n}$. Denote by $\pi$ the projection
$\mathbb{R}^{n}\times\mathbb{R}\to\mathbb{R}^{n}$. Then (for $D_{X}$
sufficiently small) the disk $\iota(D_{X})$ is a smooth graph over its
projection. We thus have that $\left.\iota\right|_{\iota^{-1}(C_{R/2})}$ is a
local diffeomorphism with $B^{n}_{R/2}(0)$. Fix a connected component of
$\iota^{-1}(C_{R/2})$, which we denote $S_{0}$. Then
$\left.\iota\right|_{S_{0}}:S_{0}\to B^{n}_{R/2}(0)$ is a local
diffeomorphism.

The condition on $\nu\cdot e_{n+1}$ guarantees that
$\left.\iota\right|_{S_{0}}$ is transverse to any line of the form
$\\{q\\}\times\mathbb{R}$ and the intersection is always positive. Moreover,
the intersection index of $M$ with such lines is constant (since $\sup_{M\cap
C_{\frac{R}{2}}}|x_{n+1}|<\frac{R}{2}$, $M\cap C_{\frac{R}{2}}$ has no
boundary in $B^{n}_{\frac{R}{2}}(0)\times\mathbb{R}$). Therefore
$\iota^{-1}(\\{q\\}\times\mathbb{R})$ is a subset of $S_{0}$ with fixed
cardinality $N\in\mathbb{N}$, regardless of $q$. (The immersion is proper,
therefore there can only be finitely many points of intersection.)

The above observations imply that $\left.\iota\right|_{S_{0}}$ is a $N$-cover
of $B^{n}_{R/2}(0)$. On the other hand, the ball $B^{n}_{R/2}(0)$ is its own
universal cover (and $S_{0}$ is connected), so $N=1$. We have proved that each
connected component of $\iota^{-1}(C_{R/2})$ is mapped (by $\iota$) to a
(smooth) graph over $B^{n}_{R/2}(0)$, which provides the smooth functions
$v_{j}$ in the conclusion of Theorem 6 (where $j$ ranges over the set of
connected components, which are finitely many because $\iota$ is proper).

At this stage, one can follow the arguments of De Giorgi [5], or directly
invoke the De Giorgi–Nash–Moser theory, to conclude that $g$ is Hölder
continuous on every $\text{graph}(v_{j})$, and that each $v_{j}$ is in
$C^{1,\alpha}(B^{n}_{R/2}(0))$, with the estimate $\|\nabla
v_{j}\|_{C^{0,\alpha}(B^{n}_{R/2}(0))}\leq C(n)E_{0}$. Higher regularity (and
the analogous estimate for the $C^{k,\alpha}$-norms) follow from Schauder
theory (using the Schoen inequality to control the $L^{2}$ norm of $A$ by the
tilt excess). ∎

###### Proof of Theorem 5.

The arguments given for the graph decomposition for Theorem 6 lead to the
conclusion that $\iota$ restricted to any connected component $S_{0}$ of $S$
is a local diffeomorphism and an $N$-cover of
$B^{n}_{R/2}(0)\setminus\pi(\Sigma)$. This relies on the observation that
$B^{n}_{R/2}(0)\setminus\pi(\Sigma)$ is open and (path) connected (a
consequence of the fact that $\Sigma$ is closed with
$\mathcal{H}^{n-1}(\Sigma)=0$, which follows from $\text{cap}_{2}(\Sigma)=0$).
This guarantees the possibility to choose a normal that has positive
intersections with lines $\\{q\\}\times\mathbb{R}$ and the constancy of the
intersection index of $M$ with such lines for $q\in
B^{n}_{R/2}(0)\setminus\pi(\Sigma)$.

At this stage, we have a description of $M\cap C_{R/2}$ as graph of a smooth
$q$-valued function on $B^{n}_{R/2}(0)\setminus\pi(\Sigma)$. For any $q\in
B^{n}_{R/2}(0)\setminus\pi(\Sigma)$, by ordering the values
$\Pi\Big{(}M\cap(\\{q\\}\times\mathbb{R})\Big{)}\subset\mathbb{R}$
increasingly, where $\Pi$ is the projection onto the second factor of
$B^{n}_{R/2}(0)\times\mathbb{R}$, we obtain $q$ Lipschitz functions
$u_{j}:B^{n}_{R/2}(0)\setminus\pi(\Sigma)\to\mathbb{R}$, with Lipschitz
constant $\frac{1}{2n}$, $u_{j}\leq u_{j+1}$ for all $j\in\\{1,\ldots,Q-1\\}$,
which we can extend (preserving the Lipschitz constant) to
$u_{j}:B^{n}_{R/2}(0)\to\mathbb{R}$. ∎

Theorem 5 gives a sheeting theorem for immersions, that are allowed to possess
a singular set of locally finite $\mathcal{H}^{n-2}$-measure (or vanishing
$2$-capacity), and that are assumed to be “close” to a hyperplane. For such
immersions, genuine branch points may arise, hence the singular set cannot be
ruled out in the conclusions.

###### Proof of Theorem 7.

Specialising Theorem 5 to embeddings, that is, if $\iota(M)$ is properly
embedded in $C_{R}\setminus\Sigma$, then the Lipschitz functions
$u_{j}:B^{n}_{R/2}(0)\setminus\pi(\Sigma)\to\mathbb{R}$ must be such that, for
every $j\in\\{1,\ldots,Q-1\\}$, $u_{j}<u_{j+1}$. Thanks to the strict
inequality, each $u_{j}$ is a Lipschitz solution of the weak minimal surface
PDE on $B^{n}_{R/2}(0)\setminus\pi(\Sigma)$, hence a smooth strong solution.
Simon’s well-known singularity removal [15], which only requires $\pi(\Sigma)$
closed in $B^{n}_{R/2}(0)$ and $\mathcal{H}^{n-1}(\pi(\Sigma))=0$ (a
consequence of $\text{cap}_{2}(\Sigma)=0$), yields a smooth extension
$u_{j}:B^{n}_{R/2}(0)\to\mathbb{R}$ for each $j$, so that $\text{sing}_{M}\cap
C_{\frac{R}{2}}=\emptyset$. ∎

######  Remark 5.1.

As shown in [13], Theorem 7 leads rather quickly to the renowned Schoen–Simon
regularity and compactness theory for stable minimal embedded hypersurfaces,
see [13, Theorems 2 and 3].

The extra step required for this is a fairly simple slicing argument, see [13,
pp. 785–787], which proves that “closeness” to a classical cone cannot arise
for embeddings; after that, standard tangent cone analysis and dimension
reduction complete the proof. For contrast, in the immersed case, closeness to
classical cones can arise (and one would naturally aim for a sheeting result,
over the several hyperplanes constituting the classical cone, which for $n\leq
6$ and in the absence of singular set follows from Lemma 3.3 of Part I).

With the multi-valued description of $M$ in Theorem 5, natural questions are a
more precise characterisation of the $q$-valued function obtained (plausibly,
one can establish $C^{1,\alpha}$ regularity in the sense of $q$-valued
functions), and a finer structure result for the singular set. While we do not
pursue this here, we observe:

######  Corollary 1 (uniqueness of tangent hyperplanes).

Let $M$ be as in the beginning of Part II, and let $x\in\overline{M}$ be such
that there exists a tangent cone (in the sense of varifolds) to $M$ at $x$
that is a hyperplane with multiplicity. Then that is the unique tangent cone
at $x$.

###### Proof.

We take a blow up that gives rise to a hyperplane with multiplicity, which we
assume to be $\\{x_{n+1}=0\\}$ by rotating coordinates. For the blow up
sequence $M_{\ell}$ (obtained by dilations of $M$) we have $E_{M_{\ell}}(1)\to
0$ (this follows from the monotonicity formula, using also Remark 4.2).
Denoting by $g_{\ell}$ the tilt function on $M_{\ell}$, using the estimate
$\sup_{M_{\ell}\cap C_{\frac{R}{2}}}g_{\ell}\leq
c(n)E_{M_{\ell}}(R)^{\alpha}$, $\alpha=\alpha(n)\in(0,1)$, obtained in Theorem
5, it follows that $\sup_{M_{\ell}\cap C_{\frac{1}{2}}}g_{\ell}\to 0$. If any
other blow up gave rise to a different cone, we would have the existence of
$y_{\ell}\in M_{\ell}\cap C_{\frac{1}{2}}$ with $y_{\ell}\to 0$ and
$\limsup_{\ell\to\infty}g_{\ell}(y_{\ell})>0$, contradiction. ∎

If $U=\mathbb{R}^{n+1}$ and the mass growth of $M$ at infinity is Euclidean,
then tangents at infinity exist and are cones. The same argument shows:

######  Corollary 2 (uniqueness of tangent hyperplanes at infinity).

Let $M$ be as in the beginning of Part II with $U=\mathbb{R}^{n+1}$, and
assume that one tangent cone to $M$ at infinity (in the sense of varifolds) is
a hyperplane with multiplicity. Then that is the unique tangent cone at
infinity.

######  Remark 5.2.

If the multiplicity of the hyperplane is at most $2$, then these uniqueness
results follow from [18].

Acknowledgments. I wish to thank Otis Chodosh and Paul Minter for fruitful and
helpful comments on the manuscript.

###  Appendix A The case $n=3$ of Theorem 3

While not essential for our arguments, we note explicitly that when $n=3$ a
stronger conclusion in Theorem 3 can be obtained from the proof given:

######  Corollary 3 ($n=3$).

Let $M$ be a properly immersed two-sided stable minimal hypersurface in
$B_{2R}(0)$, with $0\in M$. There exists an (explicit) increasing continuous
function $y:[0,\infty)\to[0,\infty)$ with $y(0)=0$, such that for every $x\in
M\cap B_{R/2}(0)$ we have

$|A|(x)\leq\frac{y\Bigg{(}\frac{1}{2R}\int_{B_{2R}}|A|^{2}\Bigg{)}}{R}.$

######  Remark A.1.

The proof gives $y(a)\sim a$ for $a$ large and $y(a)\sim\sqrt{a}$ for $a$
small.

###### Proof.

Repeating the proof of Theorem 3 with $n=3$ until the choice of $d$, and
noting that
$\frac{1}{R^{2}d^{\frac{4}{n-2}}}+d^{\frac{2(n-4)}{n-2}}=\frac{1}{R^{2}d^{4}}+\frac{1}{d^{2}}$,
if we let $d=\frac{x}{R}$ the decay relation becomes

$S_{\ell+1}\leq
c\Big{(}\frac{1}{x^{4}}+\frac{1}{x^{2}}\Big{)}^{3}R^{6}\tilde{C}^{\ell}S_{\ell-1}^{3},$

with $\tilde{C}=2^{21}$ and $c=3^{3}\cdot 2^{48}$ (using rough estimates,
among which $C_{MS}^{3}\leq 4^{28}$). A sufficient smallness condition on
$S_{1}$ (to have $S_{\ell}\to 0$) is then

$R^{3}S_{1}=R^{3}\int_{M\cap
B_{R}}|A|^{6}\leq\Big{(}\frac{x^{4}}{1+x^{2}}\Big{)}^{\frac{3}{2}}\frac{1}{c^{\frac{1}{2}}\tilde{C}^{\frac{3}{2}}}.$

This is in turn implied444We use
$R^{3}\int_{B_{R}}|A|^{6}\leq(40\,C_{MS})^{3}\big{(}\frac{1}{2R}\int_{B_{2R}\cap
M}|A|^{2}\big{)}^{3}$, obtained in (11)., writing
$K=c^{\frac{1}{3}}\tilde{C}(40\,C_{MS})^{2}$, by

$K\Bigg{(}\frac{1}{2R}\int_{M\cap
B_{2R}}|A|^{2}\Bigg{)}^{2}\leq\frac{x^{4}}{1+x^{2}}.$

As $\frac{x^{4}}{1+x^{2}}$ is monotonically (strictly) increasing with value
$0$ at $0$, we let $f$ denote its inverse function and set $y(a)=f(Ka^{2})$.
Then by choosing $d=\frac{y(a)}{R}$, with
$a=\frac{1}{2R}\int_{B_{2R}}|A|^{2}$, we find $|A|\leq\frac{y(a)}{R}$ on
$B_{\frac{R}{2}}$. ∎

######  Remark A.2.

In other words, for $n=3$ curvature estimates of Theorem 2 completely follow
from Corollary 3 (without appealing to tangent cone analysis and dimension
reducing). Indeed,
$\frac{1}{(2R)^{n-2}}\int_{B_{2R}}|A|^{2}\leq\omega_{n}2^{n}\Lambda$ (by the
stability inequality in $B_{4R}$), hence
$|A|(x)\leq\frac{y(8\omega_{3}\Lambda)}{R}$ for every $x\in B_{\frac{R}{2}}$.

More precisely, as we have an explicit $y$ from the above proof, for $n=3$ the
above result yields $\beta$ in Theorem 2 as a constant explicitly computable
in terms of $\Lambda$, namely
$\beta=\sqrt{\frac{\sigma+\sqrt{\sigma^{2}+4\sigma}}{2}}$ with
$\sigma=K(8\omega_{3}\Lambda)^{2}$ and $K$ as above. Combining this with [4],
which gives an absolute explicit upper bound for the mass of $M\cap B_{2R}$
under the hypotheses of Theorem 2, one obtains an explicit upper bound (an
absolute constant, independent of $\Lambda$) for $\sup_{B_{\frac{R}{4}}}R|A|$
in Theorem 2 when $n=3$.

###  Appendix B An elementary lemma

######  Lemma B.1.

Let $\tilde{C},C>0,\alpha>0$ be given constants, and let $x_{\ell}$ be a
sequence of positive real numbers that satisfies the following recursive
relation for all $\ell\in\mathbb{N}\setminus\\{0\\}$:

$x_{\ell+1}\leq\tilde{C}\,C^{\ell}\,x_{\ell}^{1+\alpha}.$

Assume that
$x_{1}\leq\frac{1}{(\tilde{C}\,C^{1+\frac{1}{\alpha}})^{\frac{1}{\alpha}}}$ if
$C>1$, $x_{1}<\frac{1}{\tilde{C}^{\frac{1}{\alpha}}}$ if $C\leq 1$. Then
$x_{\ell}\to 0$ as $\ell\to\infty$.

###### Proof.

Assume that $C>1$. We show that there exists $a\in(0,1)$ such that
$\tilde{C}\,C^{\ell}\,x_{\ell}^{\alpha}\leq a$ for all $\ell\in\mathbb{N}$,
from which $x_{\ell+1}\leq ax_{\ell}$ follows (hence the conclusion). For
$\ell=1$ we have

$\tilde{C}Cx_{1}^{\alpha}\leq\frac{C\,\tilde{C}}{\tilde{C}\,C^{1+\frac{1}{\alpha}}}=\frac{1}{C^{\frac{1}{\alpha}}}$

and we set $a=\frac{1}{C^{\frac{1}{\alpha}}}$. Now we check inductively, for
arbitrary $(\ell+1)\geq 2$, that

$\tilde{C}\,C^{\ell+1}\,x_{\ell+1}^{\alpha}\leq\tilde{C}\,C^{\ell+1}(\tilde{C}\,C^{\ell}\,x_{\ell}^{1+\alpha})^{\alpha}=C(\tilde{C}C^{\ell}x_{\ell}^{\alpha})^{1+\alpha}\leq
Ca^{1+\alpha}=\frac{C}{C^{\frac{1+\alpha}{\alpha}}}=a.$

If $C\leq 1$ then the recursive relation implies
$x_{\ell+1}\leq\tilde{C}\,x_{\ell}^{1+\alpha}\leq(\tilde{C}\,x_{\ell}^{\alpha})x_{\ell}$,
in which case the smallness assumption
$x_{1}<\frac{1}{\tilde{C}^{\frac{1}{\alpha}}}$ implies the conclusion. ∎

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