$\mathbf{C}^{\prime\prime}\in\mathcal{C}$ and $m^{\prime\prime}\in\mathbb{N}$
satisfying $m^{\prime\prime}\theta_{\mathbf{C}^{\prime\prime}}(0)\leq
m\theta_{\mathbf{C}}(0)\leq\tilde{\theta}_{l}$. We can assume that
$m^{\prime\prime}[\mathbf{C}^{\prime\prime}]\llcorner(A_{\infty,0},g_{\mathrm{eucl}})$
is a $(\mathbf{C},m,\beta)$-strong-cone region, and hence

$d_{H}(\mathbf{C}^{\prime\prime}\cap B_{1},\mathbf{C}\cap
B_{1})\leq\beta,\quad
m^{\prime\prime}\theta_{\mathbf{C}^{\prime\prime}}(0)\geq
m\theta_{\mathbf{C}}(0)-\beta.$

Ensuring $\beta(l)$ is sufficiently small, Theorem 5.1 implies
$\mathbf{C}^{\prime\prime}\in\mathcal{C}(\mathbf{C})$, and hence
$\theta_{\mathbf{C}^{\prime\prime}}(0)=\theta_{\mathbf{C}}(0)$ and
$m^{\prime\prime}=m$. This proves my claim.

We break into two subcases.

Subcase 1A: $\theta_{V^{\prime}}(a)\geq\tilde{\theta}_{l}$ for some
$a\in\mathrm{spt}V^{\prime}$. In this case, by the monotonicity formula we
must have $V^{\prime}=m[a+\mathbf{C}^{\prime}]$ for some
$\mathbf{C}^{\prime}\in\mathcal{C}(\mathbf{C})$. If $\mathbf{C}$ is planar,
there is no loss in assuming $a\in\mathbf{C}^{\perp}$.

Since each $V_{i}^{\prime}\llcorner(A_{4,1}(0),g_{i}^{\prime})$ is a
$(\mathbf{C},m,\beta^{\prime\prime},\tau,\sigma)$-weak-cone region, for every
$r\in[1,4]$ there are $a_{r,i}\equiv a_{r}(V_{i}^{\prime})\in\mathbb{R}^{8}$,
$\mathbf{C}_{r,i}\in\mathcal{C}(\mathbf{C})$, and $C^{2}$ functions
$\\{u_{r,i,j}:(a_{r,i}+\mathbf{C}_{r,i})\cap
A_{r,r/8}(a_{r,i})\to\mathbf{C}_{r,i}^{\perp}\\}_{j=1}^{m}$ such that

(69) $V_{i}^{\prime}\llcorner
A_{r,r/8}(a_{r,i})=\sum_{j=1}^{m}[G_{a_{r,i}+\mathbf{C}_{r,i}}(u_{r,i})\cap
A_{r,r/8}(a_{r,i})]_{g_{i}^{\prime}},$

and

(70) $r^{-1}|u_{r,i,j}|+|\nabla
u_{r,i,j}|+r|\nabla^{2}u_{r,i,j}|\leq\beta^{\prime\prime}\quad\forall i,j,r.$

Moreover, by construction we have $a_{1,i}=0$.

It follows by (69), (70), Ahlfors regularity (19), and varifold convergence
$V_{i}^{\prime}\to[a+\mathbf{C}^{\prime}]$, that for every $r\in[1,4]$ and
$i>>1$, we have

$d_{H}((a+\mathbf{C}^{\prime})\cap
A_{r/2,r/4}(a_{r,i}),(a_{r,i}+\mathbf{C}_{r,i})\cap A_{r/2,r/4}(a_{r,i}))\leq
8\beta^{\prime\prime}.$

(Note the $\beta^{\prime\prime}$ instead of $\beta^{\prime}$!) By Lemma 5.3
(applied in the ball $B_{r/2}(a_{r,i})$), ensuring
$\beta^{\prime\prime}(l,\tau)$ is sufficiently small, we deduce that

(71) $|a|\leq 2^{-10}\tau\leq 2^{-4},\quad|a-a_{r,i}|\leq 2^{-10}\tau r,\quad
d_{H}(\mathbf{C}^{\prime},\mathbf{C}_{r,i})\leq 2^{-10}\tau,$

for each $r\in[1,4]$, and all $i>>1$.

I claim that $\mathcal{I}^{\prime}\not\subset\\{a\\}$ if
$\theta_{\mathbf{C}^{\prime}}(0)>1$, and $\\#\mathcal{I}^{\prime}\geq 2$ if
$\theta_{\mathbf{C}^{\prime}}(0)=1$. Let us prove this. If
$\theta_{\mathbf{C}}(0)>1$ and $\mathcal{I}^{\prime}\subset\\{a\\}$ then set
$a^{\prime}=a$ and $r^{\prime}=1/2$. If, on the other hand,
$\theta_{\mathbf{C}}(0)=1$ and
$\mathcal{I}^{\prime}=\\{y^{\prime}\\}\subset\overline{B_{1/8}}\cap(a+\mathbf{C}^{\prime})$,
then set $a^{\prime}=a+2^{-4}\tau(y^{\prime}-a)$, $r^{\prime}=1-2^{-5}\tau<1$.
In this case we get

(72) $|a^{\prime}-y^{\prime}|<r^{\prime}/8,\quad|a^{\prime}-a|\leq
2^{-6}\tau,\quad|a^{\prime}-a_{r,i}|\leq 2^{-5}\tau.$

Either way $a^{\prime}+\mathbf{C}^{\prime}=a+\mathbf{C}^{\prime}$.

In either case, by assumption and our choice of $a^{\prime},r^{\prime}$, we
can find for all $i>>1$ $C^{2}$ functions
$\\{u_{i,j}^{\prime}:(a+\mathbf{C}^{\prime})\cap
A_{1,r^{\prime}/8}(a^{\prime})\to\mathbf{C}^{\prime\perp}\\}_{j=1}^{m}$ with
the properties that

(73) $V_{i}^{\prime}\llcorner
A_{1,r^{\prime}/8}(a^{\prime})=\sum_{j=1}^{m}[G_{a^{\prime}+\mathbf{C}^{\prime}}(u_{i,j}^{\prime})\cap
A_{1,r^{\prime}/8}(a^{\prime})]_{g_{i}^{\prime}},$

and

(74) $|u_{i,j}^{\prime}|+|\nabla
u_{i,j}^{\prime}|+|\nabla^{2}u_{i,j}^{\prime}|\leq\beta^{\prime\prime}/2.$

On the other hand, since $V_{i}^{\prime}\to[a^{\prime}+\mathbf{C}^{\prime}]$
as varifolds and $\theta_{\mathbf{C}^{\prime}}(0)=\theta_{\mathbf{C}}(0)$, we
have
$|\theta_{V_{i}^{\prime}}(a^{\prime},r)-m\theta_{\mathbf{C}}(0)|\leq\beta^{\prime\prime}$
for all $i>>1$ and $r\in[1/2,4]$. This, together with (69), (70), (72), (73),
(74), implies that
$V_{i}^{\prime}\llcorner(A_{4,r^{\prime}}(0),g_{i}^{\prime})$ is a
$(\mathbf{C},m,\beta^{\prime\prime},\tau,\sigma)$-weak-cone region,
contradicting (68). This proves my claim.

If $\theta_{\mathbf{C}^{\prime}}(0)>1$, we have
$\mathcal{I}^{\prime}\not\subset\\{a\\}$, and $\mathcal{I}^{\prime}\subset
B_{1/4}(a)$, and so we can apply Case 2 of $(\dagger_{l,I})$ to the
translated/dilated sequences $\eta_{a,1/4}(V_{i}^{\prime})$,
$g_{i}^{\prime}\circ\eta_{a,1/4}^{-1}$. We obtain a finite collection
$\mathcal{S}^{\prime}$ of $(\tilde{\theta}_{l},\sigma,\beta)$-models, an
integer $N^{\prime}$, and a radii
$r^{\prime\prime}_{i}\in(\frac{4}{5}\cdot\frac{1}{4},\frac{1}{4})$ (recall our
assumption on $\sigma$), so that every
$V_{i}^{\prime}\llcorner(B_{r^{\prime\prime}_{i}}(a),g_{i}^{\prime})$ (for
$i>>1$) admits a
$(\tilde{\theta}_{l},\beta,\mathcal{S}^{\prime},N^{\prime})$-strong-cone
decomposition.

If $\theta_{\mathbf{C}^{\prime}}(0)=1$, we have $\\#\mathcal{I}^{\prime}\geq
2$ and $\mathcal{I}^{\prime}\subset B_{1/4}(a)$, and so we can apply Case 3 of
$(\dagger_{l,I})$ as in the previous paragraph to deduce the same conclusion.

By Lemma 5.4 and (71), for $\tau(\gamma,l)$ sufficiently small, we can write

$(a+\mathbf{C}^{\prime})\cap
A_{\infty,1/2}=G_{\mathbf{C}^{\prime}}(u^{\prime})\cap A_{\infty,1/2}$

for $u^{\prime}:\mathbf{C}^{\prime}\cap
A_{\infty,1/2}\to\mathbf{C}^{\prime\perp}$ a $C^{2}$ function satisfying
$|x|^{-1}|u^{\prime}|+|\nabla
u^{\prime}|+|x||\nabla^{2}u^{\prime}|\leq\gamma$. Letting
$S^{\prime}=m[a+\mathbf{C}^{\prime}]$, then this and our requirement $|a|\leq
1/16$ imply $(S^{\prime},\mathbf{C},\\{(\mathbf{C},B_{1/4}(a))\\})$ is a
$(\tilde{\theta}_{l},\sigma,\gamma)$-smooth model. $C^{2}$ convergence to
$m[a+\mathbf{C}^{\prime}]$ means that for $i>>1$, every
$V_{i}^{\prime}\llcorner(B_{1},g_{i}^{\prime})$ is a
$(S^{\prime},\beta)$-smooth region.

Combining the above two paragraphs, we get that every
$V_{i}^{\prime}\llcorner(B_{1},g_{i}^{\prime})$, and hence
$V_{i}\llcorner(B_{\rho_{i}}(a_{i}),g_{i})$, admits a
$(\tilde{\theta}_{l},\beta,\\{S^{\prime}\\}\cup\mathcal{S}^{\prime},1+N^{\prime})$-strong-
cone decomposition. If we let $S$ be the smooth model as constructed in the
$\rho_{i}=0$ case, then like we argued before we deduce that
$V_{i}\llcorner(B_{1-5\sigma}(0),g_{i})$ admits a
$(\tilde{\theta}_{l},\beta,\\{S,S^{\prime}\\}\cup\mathcal{S}^{\prime},3+N^{\prime})$-strong-
cone decomposition. This is a contradiction, and completes the proof of Case
1A.

Subcase 1B: $\theta_{V^{\prime}}(x)\leq\tilde{\theta}_{l-1}$ for all
$x\in\mathrm{spt}V^{\prime}.$ Write
$\mathrm{sing}V^{\prime}\cup\mathcal{I}^{\prime}=\\{y_{1}^{\prime},\ldots,y_{d^{\prime}}^{\prime}\\}$.
By Lemma 4.1 and our subcase hypothesis, for each $\alpha=1,\ldots,d^{\prime}$
there is a tangent cone to $V^{\prime}$ at $y_{\alpha}^{\prime}$ of the form
$m_{\alpha}[\mathbf{C}_{\alpha}^{\prime}]$, where
$\mathbf{C}_{\alpha}^{\prime}\in\mathcal{C}$ and
$m_{\alpha}\theta_{\mathbf{C}_{\alpha}^{\prime}}(0)\leq\tilde{\theta}_{l-1}$.
By Lemmas 4.1, 4.4, Theorem 6.3, Remark 6.4, and the maximum principle of
[Ilm96], we can choose radii $r_{\alpha}$ so that the following holds:

  1. (1)

the collection $\\{B_{2r_{\alpha}}(y_{\alpha}^{\prime})\\}_{\alpha}\subset
B_{1/2}(0)$ are disjiont;

  2. (2)

for each $\alpha$,
$\mathrm{spt}V^{\prime}\cap(A_{2r_{\alpha},0}(y_{\alpha}),g_{\mathrm{eucl}})$
is a $(\mathbf{C}_{\alpha}^{\prime},1,\gamma)$-strong-cone region;

  3. (3)

for each $\alpha$, we have

$d_{H}(\mathrm{spt}V^{\prime}\cap
B_{r_{\alpha}}(y_{\alpha}^{\prime}),(y_{\alpha}^{\prime}+\mathbf{C}_{\alpha}^{\prime})\cap
B_{r_{\alpha}}(y_{\alpha}^{\prime})\leq\delta_{l-1,I}/2,$

and

$(m_{\alpha}-1/4)\theta_{\mathbf{C}_{\alpha}^{\prime}}(0)\leq\theta_{V^{\prime}}(y_{\alpha},r_{\alpha}/2),\quad\theta_{V^{\prime}}(y_{\alpha},r_{\alpha})\leq(m+1/4)\theta_{\mathbf{C}_{\alpha}^{\prime}}(0)$

Here $\delta_{l-1,I}=\delta_{l-1,I}(l-1,I,\beta,\sigma)$ is the constant from
$(\dagger_{l-1,I})$.

For each $\alpha$, we can therefore apply $(\dagger_{l-1,I})$ to the sequences
$(\eta_{y_{\alpha},r_{\alpha}})_{\sharp}V_{i}^{\prime}$,
$g_{i}^{\prime}\circ\eta_{y_{\alpha},r_{\alpha}}^{-1}$. We obtain a finite
collection $\mathcal{S}^{\prime}$ of
$(\tilde{\theta}_{l},\sigma,\beta)$-smooth models, an integer $N^{\prime}$,
and radii $r_{\alpha,i}^{\prime}\in(\frac{4}{5}r_{\alpha},r_{\alpha})$, so
that each
$V_{i}^{\prime}\llcorner(B_{r_{\alpha,i}^{\prime}}(y_{\alpha}^{\prime}),g_{i}^{\prime})$
admits a $(\tilde{\theta}_{l},\beta,\mathcal{S}^{\prime},N^{\prime})$-strong-
cone decomposition, for $i>>1$.

Recall that $V^{\prime}\llcorner(A_{\infty,1},g_{\mathrm{eucl}})$ is a
$(\mathbf{C},m,\beta)$-strong-cone region, and $\beta\leq\gamma$. By our
choice of $\alpha$, we get that
$(S=V^{\prime},\mathbf{C}^{\prime},\\{(\mathbf{C}_{\alpha}^{\prime},B_{r_{\alpha}}(y_{\alpha}^{\prime}))\\}_{\alpha})$
is a $(\tilde{\theta}_{l},\sigma,\gamma)$-smooth model. $C^{2}$ convergence
implies $V_{i}^{\prime}\llcorner(B_{1},g_{i}^{\prime})$ is a
$(S,\beta)$-smooth region provided $i>>1$.

It follows that each $V_{i}^{\prime}\llcorner(B_{1},g_{i}^{\prime})$ admits a
$(\tilde{\theta}_{l},\beta,\\{S\\}\cup\mathcal{S}^{\prime},1+N^{\prime})$-strong-
cone decomposition. The proof of Case 1B now continues as in Case 1A.

Case 2: $\mathcal{I}\not\subset\\{0\\}$ and $\theta_{\mathbf{C}}(0)>1$. Let
$\mathcal{I}\setminus\\{0\\}=\\{y_{1},\ldots,y_{d^{\prime}}\\}$, which is non-
empty by assumption. We can choose a radius $r\in(1-20\sigma(I+1),1-5\sigma)$
with the property that
$\max\\{||y_{\alpha}|-r||:y_{\alpha}\in\mathcal{I}\\}\geq 5\sigma$.

Set $r_{0}=\min\\{|y_{\alpha}|:y_{\alpha}\in\mathcal{I}\\}/4$, and $y_{0}=0$,
and then for each $\alpha=1,\ldots,d^{\prime}$ with $|y_{\alpha}|<r$, choose a
radius $r_{\alpha}$ so that:

  1. (1)

the collection $\\{B_{2r_{\alpha}}(y_{\alpha})\\}_{\alpha}\subset
A_{r-4\sigma,2r_{0}}(0)$ are disjoint;

  2. (2)

for each $\alpha$, if we set $\mathbf{C}_{\alpha}=T_{y_{\alpha}}\mathbf{C}$ to
be the tangent plane of $\mathbf{C}$, then
$\mathbf{C}\cap(A_{2r_{\alpha},0}(y_{\alpha}),g_{\mathrm{eucl}})$ is a
$(\mathbf{C}_{\alpha},1,\gamma)$-strong-cone region;

  3. (3)

for each $\alpha$, we have

$d_{H}(\mathbf{C}\cap
B_{r_{\alpha}}(y_{\alpha}),(y_{\alpha}+\mathbf{C}_{\alpha})\cap
B_{r_{\alpha}}(y_{\alpha}))\leq\delta_{l-1,I}/2,$

and

$m-1/4\leq\theta_{\mathbf{C}}(y_{\alpha},r_{\alpha}/2),\quad\theta_{\mathbf{C}}(y_{\alpha},r_{\alpha})\leq
m+1/4.$

Note that every $m\theta_{\mathbf{C}_{\alpha}}(0)=m<m\theta_{\mathbf{C}}(0)$,
and so in fact $m\theta_{\mathbf{C}_{\alpha}}(0)\leq\tilde{\theta}_{l-1}$.

For $i>>1$ we can apply $(\dagger_{l-1,I})$ to find a finite collection
$\mathcal{S}^{\prime}$ of $(\tilde{\theta}_{l},\sigma,\beta)$-smooth models,
an integer $N^{\prime}$, and radii
$r_{\alpha,i}^{\prime}\in(\frac{4}{5}r_{\alpha},r_{\alpha})$, so that every
$V_{i}\llcorner(B_{r^{\prime}_{\alpha,i}}(y_{\alpha}),g_{i})$ admits a
$(\tilde{\theta}_{l},\beta,\mathcal{S}^{\prime},N^{\prime})$-strong-cone
decomposition.

On the other hand, since $\mathrm{index}(V_{i},B_{r_{0}}(0),g_{i})\leq I-1$
for all $i>>1$, we can apply $(\dagger_{l,I-1})$ to find another finite
collection $\mathcal{S}^{\prime\prime}$, an integer
$N^{\prime\prime}\in\mathbb{N}$, and radii
$r^{\prime}_{0,i}\in(\frac{4}{5}r_{0},r_{0})$, so that each
$V_{i}\llcorner(B_{r^{\prime}_{0,i}}(0),g_{i})$ admits a
$(\tilde{\theta}_{l},\beta,\mathcal{S}^{\prime\prime},N^{\prime\prime})$-strong-
cone decomposition.

Now observe that
$(m[\mathbf{C}],\mathbf{C},\\{(\mathbf{C}_{\alpha},B_{\frac{r_{\alpha}}{r}}(\frac{y_{\alpha}}{r}))\\}_{\alpha=0}^{d^{\prime}})$
is a $(m\theta_{\mathbf{C}}(0),\sigma,\gamma)$-smooth model, and each
$V_{i}\llcorner(B_{r},g_{i})$ is a $(S,\beta)$-smooth region for $i>>1$.

Combining all of the above, if we take
$\mathcal{S}=\\{S\\}\cup\mathcal{S}^{\prime}\cup\mathcal{S}^{\prime\prime}$,
$N=IN^{\prime}+N^{\prime\prime}+1$, then every $V_{i}\llcorner(B_{r},g_{i})$
admits a $(\tilde{\theta}_{l},\beta,\mathcal{S},N)$-strong-cone decomposition.
This is a contradiction, and finishes the proof of Case 2.

Case 3: $\mathcal{I}\not\subset\\{0\\}$ and $\theta_{\mathbf{C}}(0)=1$. In
this case, since $\mathrm{sing}\mathbf{C}=\emptyset$, $V_{i}\to m[\mathbf{C}]$
in $C^{2}$ on compact subsets of $B_{1}\setminus\mathcal{I}$. Pick a radius
$r\in(1-20\sigma(I+1),1-5\sigma)$ with the property that
$\max\\{||y_{\alpha}|-r||:y_{\alpha}\in\mathcal{I}\\}\geq 5\sigma$. Let
$\mathcal{I}^{\prime}=\mathcal{I}\cap B_{r}=\\{1,\ldots,d^{\prime}\\}$.

For each $y_{\alpha}\in\mathcal{I}^{\prime}$, choose a radius $r_{\alpha}$ so
that all the $\\{B_{2r_{\alpha}}(y_{\alpha})\\}_{\alpha}\subset B_{r-4\sigma}$
are disjoint. Define the rescaled varifolds
$V^{\alpha}_{i}=(\eta_{y_{\alpha},r_{\alpha}})_{\sharp}V_{i}$ and metrics
$g_{i}^{\alpha}=g_{i}\circ\eta_{y_{\alpha},r_{\alpha}}^{-1}$. Then for each
$\alpha,i$ we have $\mathrm{index}(V^{\alpha}_{i},B_{1},g_{i}^{\alpha})\leq
I$, and $V^{\alpha}_{i}$ converge to $m[\mathbf{C}]$ as varifolds in $B_{1}$,
and converge to $\mathbf{C}$ in $C^{2}$ on compact subsets of
$B_{1}\setminus\\{0\\}$. We can therefore apply Case 1 of $(\dagger_{l,I})$ to
each sequence to deduce the existence of a finite collection
$\mathcal{S}^{\prime}$ of $(\tilde{\theta}_{l},\sigma,\beta)$-smooth models,
and an integer $N^{\prime}$, and radii
$r^{\prime}_{\alpha,i}\in(\frac{4}{5}r_{\alpha},r_{\alpha})$, so that every
$V_{i}\llcorner(B_{r^{\prime}_{\alpha,i}}(y_{\alpha}),g_{i})$ admits a
$(\tilde{\theta}_{l},\beta,\mathcal{S}^{\prime},N^{\prime})$-strong-
decomposition whenever $i>>1$.

An important note: if $\\#\mathcal{I}\geq 2$, then by Lemma 4.4 and our choice
of balls $B_{r_{\alpha}}(y_{\alpha})$, we have
$\mathrm{index}(V^{\alpha}_{i},B_{1},g_{i}^{\alpha})\leq I-1$ for each
$\alpha=1,\ldots,d^{\prime}$ and $i>>1$. We can therefore apply Case 1 of
$(\dagger_{I-1,l})$ to the sequence $V^{\alpha}_{i}$ rather than Case 1 of
$(\dagger_{l,I})$, to deduce the same result.

Now
$(S=m[\mathbf{C}],\mathbf{C},\\{(\mathbf{C}_{\alpha}=\mathbf{C},B_{\frac{r_{\alpha}}{r}}(\frac{y_{\alpha}}{r}))\\}_{\alpha=1}^{d^{\prime}})$
is a $(m,\sigma,0)$-smooth model. By $C^{2}$ convergence of $V_{i}$ to
$m[\mathbf{C}]$ away from $\mathcal{I}$, and our choice of $r$, the
$V_{i}\llcorner(B_{r},g_{i})$ are $(S,\beta)$-smooth regions for $i>>1$.

Therefore, we deduce that $V_{i}\llcorner(B_{r}(0),g_{i})$ admits a
$(\tilde{\theta}_{l},\beta,\\{S\\}\cup\mathcal{S}^{\prime},IN^{\prime}+1)$-strong-
cone decomposition for all $i>>1$. This is a contradiction, and completes the
proof of Case 3. ∎

##  8\. Parameterization

In this section we construct “parameterizations” for our cone regions. We show
that every $M$ admitting a cone-decomposition can be realized as the image
under some Lipschitz map of one of finitely-many $\\{M_{v}\\}_{v}$, depending
only on the parameters of the decomposition. In this section we work in
$\mathbb{R}^{8}$.

This section is annoying technical, due almost entirely to multiplicity. If
all our $M$s were single-sheeted (e.g. as when $M$ is an area-minimizing
boundary), then all our parameterizing maps would be $C^{1}$ perturbations of
the identity, and this section would be more or less trivial. Unfortunately,
maps $\phi$ taking multi-sheeted $M_{1}$ to $M_{2}$ are only $C^{0}$
perturbations of the identity, with the $C^{1}$ norm depending on the relative
gaps between in $M_{1}$ versus $M_{2}$. So to glue various parameterizations
together we cannot use “soft” methods, but must rely much more explicitly on
the exact structure of these maps.

We cannot get a priori control on $D\phi^{-1}$ in Theorem 8.1 for two reasons.
Intuitively: the first is that within a “class” of $M$ having the same kind of
cone decomposition, the inner radius of cone regions can be made arbitrarily
small, but the outer radius has an upper bound; the second is that sheets can
be made arbitrarily close together, but not arbitrarily far apart. For an
example of the first behavior, consider scalings of a leaf of the Hardt-Simon
foliation associated to an area-minimizing hypercone. For an example of the
second, simply consider two planes converging to each other.

Getting global regularity of $\phi$ better than Lipschitz seems to be more or
less equivalent to knowing that all the cones you see are integrable through
rotations. Even knowing this, getting better a priori bounds on $\phi$
requires sharper decay estimates in Theorem 6.3. In certain circumstances this
is known. See Remark 8.2.

It will be convenient to introduce some further notation. For each
$\mathbf{C}\in\mathcal{C}_{\Lambda}$ fix a choice of unit normal
$\nu_{\mathbf{C}}(x)$, and write
$B^{*}_{\epsilon}(\mathbf{C})=\\{x\in\mathbb{R}^{8}:\mathrm{dist}(x,\mathbf{C})<\epsilon|x|\\}$
for the conical $\epsilon$-neighborhood around $\mathbf{C}$. Let us abuse
notation slightly, and define the function
$G_{\mathbf{C}}:\mathbf{C}\times\mathbb{R}\to\mathbb{R}^{8}$ by

$G_{\mathbf{C}}(x,t)=\frac{x+t\nu_{\mathbf{C}}(x)}{|x+t\nu_{\mathbf{C}}(x)|}|x|.$

By Theorem 5.1, after shrinking $\epsilon_{0}(\Lambda)$ as necessary, if
$U_{\mathbf{C}}=\\{(x,t)\in\mathbf{C}\times\mathbb{R}:|t|<2\epsilon_{0}|x|\\}$,
then $G_{\mathbf{C}}$ restricts to a smooth diffeomorphism from
$U_{\mathbf{C}}$ onto its image, and we have $|G_{\mathbf{C}}(x,t)|=|x|$, and

(75)
$\displaystyle|G_{\mathbf{C}}(x,t)-(x+t\nu_{\mathbf{C}}(x))|\leq|t|^{2},\quad|DG_{\mathbf{C}}|_{(x,t)}-Id|\leq|t|/\epsilon_{0},$
(76) $\displaystyle(1-|x|^{-2}t^{2})|t|\leq
d(G_{\mathbf{C}}(x,t),\mathbf{C})\leq|t|,$

for all $(x,t)\in U_{\mathbf{C}}$. In the second inequality we identify
$T_{x}\mathbf{C}\times\mathbb{R}$ with $\mathbb{R}^{8}$ via
$(v,t)\leftrightarrow v+t\nu_{\mathbf{C}}(x)$.

Let $X_{\mathbf{C}}$ be the vector field in $G(U_{\mathbf{C}})\supset
B^{*}_{\epsilon_{0}}(\mathbf{C})$ defined by
$X_{\mathbf{C}}=DG_{\mathbf{C}}(\partial_{t})$, so that
$y(t)=G_{\mathbf{C}}(x,t)$ is a solution to the ODE:
$\dot{y}(t)=X_{\mathbf{C}}(y(t))$, $y(0)=x$. After shrinking
$\epsilon_{0}(\Lambda)$ possibly further we can assume that for $x\in
B_{\epsilon_{0}}^{*}(\mathbf{C})$, we have

(77) $\displaystyle X_{\mathbf{C}}(x)\perp x,\quad
0.99\leq|X_{\mathbf{C}}|\leq
1,\quad|x||DX_{\mathbf{C}}|+|x|^{2}|D^{2}X_{\mathbf{C}}|\leq 1/\epsilon_{0}.$

Let us henceforth fix $\epsilon_{0}(\Lambda)$ so that the previous discussion
holds, for any $\mathbf{C}\in\mathcal{C}_{\Lambda}$.

Given a smooth vector field $X$ defined on some open set $U$, we let
$F_{X}(x,t)$ be the flow of $X$, i.e. the map defined by
$\partial_{t}F_{X}(x,t)=X(F_{X}(x,t))$, $F_{X}(x,0)=x$. We say a mapping
$\phi:U\to U$ preserves the flowlines of $X$ if $\phi(x)=F_{X}(x,t_{\phi}(x))$
for some real-valued function $t_{\phi}$.

Given $R>\rho>0$, $a\in\mathbb{R}^{8}$, $\epsilon\leq\epsilon_{0}$, we say
$\phi|_{A_{R,\rho}(a)}$ is a $(\mathbf{C},\epsilon)$-map if $\phi$ is a
$C^{1}$ diffeomorphism $A_{R,\rho}(a)\to A_{R,\rho}(a)$, $\phi=id$ outside
$a+B^{*}_{\epsilon}(\mathbf{C})$, and $\phi$ preserves the flowlines of
$X_{\mathbf{C}}(\cdot-a)$.

Below is our main theorem of this section. We remind the reader that all
distances, derivatives, inner products, etc. are taken with respect to
$g_{\mathrm{eucl}}$ unless explicitly stated otherwise.

######  Theorem 8.1 (Parameterizing cone decompositions).

Given $\Lambda,\epsilon>0,\sigma\in(0,10^{-2}]$, there are constants
$\beta(\Lambda,\sigma,\epsilon)$, $\gamma(\Lambda,\sigma,\epsilon)$ so that
the following holds. Take $\mathcal{S}$ be any finite collection of
$(\Lambda,\sigma,\gamma)$-smooth models, $N\in\mathbb{N}$, and $\mathcal{G}$
any set of $C^{3}$ metrics on $B_{1}$. Then we can find finite collections
$\\{g_{v}\\}_{v}\subset\mathcal{G}$,
$\\{M_{v}\in\mathcal{M}_{7}(B_{1},g_{v})\\}_{v}$,
$\\{\mathbf{C}_{v}\\}_{v}\subset\mathcal{C}_{\Lambda}$, and an increasing
function $C_{\mathcal{S},N,\mathcal{G}}:[0,1)\to\mathbb{R}$, with the
properties

$\mathrm{sing}M_{v}\subset B_{1-\sigma},\quad d_{H}(M_{v}\cap
A_{1,1-\sigma},\mathbf{C}_{v}\cap A_{1,1-\sigma})\leq\epsilon,$

and so that: given any $g\in\mathcal{G}$ and $M\in\mathcal{M}_{7}(B_{1},g)$
with $M\cap B_{1}$ admitting a $(\Lambda,\beta,\mathcal{S},N)$-strong-cone
decomposition, then we can find a $v$ and a local bi-Lipschitz map
$\phi:B_{1}\to B_{1}$ such that the following holds true:

  1. (1)

$\phi(\overline{M_{v}}\cap B_{1})=\overline{M}\cap B_{1}$,
$\phi(\mathrm{sing}M_{v}\cap B_{1})=\mathrm{sing}M\cap B_{1}$;

  2. (2)

$\phi|_{B_{1}\setminus\mathrm{sing}M_{v}}$ is a $C^{2}$ diffeomorphism onto
its image;

  3. (3)

$\mathrm{Lip}(\phi|_{B_{r}})\leq C_{\mathcal{S},N,\mathcal{G}}(r)$ for any
$r<1$;

  4. (4)

$\phi|_{A_{1,1-\sigma}}$ is a $(\mathbf{C}_{v},\epsilon)$-map;

  5. (5)

$\mathrm{sing}M\subset B_{1-\sigma}$ and $d_{H}(M\cap
A_{1,1-\sigma},\mathbf{C}_{v}\cap A_{1,1-\sigma})\leq\epsilon$.

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

######  Remark 8.2.

If all the singularities of $M$ are modelled on cones of a certain type, for
example quadratic cones or other special cones being “integrable through
rotations,” then one can arrange $\phi$ in Theorem 8.1 to be globally
$C^{1,\alpha}$, though without any a priori $C^{1,\alpha}$ estimates. When
$\mathbf{C}$ is planar (as in [Whi18]), or when $\mathbf{C}$ is a minimizing
quadratic cone in Euclidean space (as in [ES19]), then I would expect one
could get $C^{1,\alpha}$ estimates on the $\phi$. See Remark 8.8.

######  Remark 8.3.

The restriction $r<1$ in Item 3 is purely technical. The reason is that the
Lipschitz norm of $\phi$ depends on distance between sheets in each $M_{v}$.
In a cone decomposition this distance is bounded away from zero inside
$B_{1}$, but could tend to $0$ as you approach $\partial B_{1}$.

Before proving Theorem 8.1 we require several technical Lemmas. The first two
construct parameterizations for a single smooth or cone region. The third is a
gluing Lemma, that allows us to glue parameterizations between two regions.

######  Lemma 8.4 (Smooth region parameterization).

Given $\Lambda,\epsilon>0$, $\sigma\in(0,10^{-2}]$, there are
$\gamma(\Lambda,\sigma,\epsilon)$, $\beta(\Lambda,\epsilon)$ so that the
following holds. Let $m\in\mathbb{N}$, and
$(S,\mathbf{C},\\{(\mathbf{C}_{\alpha},B_{r_{\alpha}}(y_{\alpha}))\\}_{\alpha})$
be a $(\Lambda,\sigma,\gamma)$-smooth model. Let
$M_{1},M_{2}\in\mathcal{M}_{7}(B_{1},g)$ for any $C^{2}$ metric $g$, and
suppose that each $M_{l}\cap(B_{1},g)$ is a $(S,\beta)$-smooth region,
$l=1,2$.

Write $O=B_{1}\setminus\cup_{\alpha}B_{r_{\alpha}/4}(y_{\alpha})$. Then we can
find a $C^{2}$ diffeomorphism $\phi:O\to O$ such that

(78) $\displaystyle\phi(M_{1})=M_{2},\quad\phi|_{A_{1,1-\sigma}}\text{ is a
$(\mathbf{C},\epsilon)$-map, }$ (79)
$\displaystyle\phi|_{A_{r_{\alpha},r_{\alpha}/4}(y_{\alpha})}\text{ is a
$(\mathbf{C}_{\alpha},\epsilon)$-map, for each $\alpha$,}$

and

(80) $\displaystyle|D\phi|_{C^{0}(O\cap B_{r})}\leq
c(S,M_{1},r)<\infty\quad\forall r<1.$

###### Proof.

Let
$\epsilon^{\prime}=\epsilon^{\prime\prime}\min\\{\epsilon_{S},\epsilon/32\\}$
for some constant
$\epsilon^{\prime\prime}(\Lambda)\leq\epsilon_{0}(\Lambda)/32$ to be
determined later, and ensure $\gamma\leq\min\\{\epsilon_{0},\epsilon\\}/32$.
Let $O^{\prime}=B_{1}\setminus\cup_{\alpha}B_{r_{\alpha}/5}(y_{\alpha})$.

Write $S\llcorner
B_{2}\setminus\cup_{\alpha}B_{r_{\alpha}/8}(y_{\alpha})=m_{1}[S_{1}]+\ldots+m_{k}[S_{k}]$
as in the Definition 7.0.1 of smooth model. Each $S_{i}$ is orientable, being
a smooth, closed (as sets) codimension-1 submanifold of the simply-connected
region $B_{2}\setminus\cup_{\alpha}B_{r_{\alpha}/8}(y_{\alpha})$. We can
therefore choose a normal vector $\nu_{i}$ on $S_{i}$ so that
$|\nu_{i}-X_{\mathbf{C}}|\leq c(\Lambda)\gamma$ on $S_{i}\cap
A_{1+\sigma,1-3\sigma}$. After replacing $X_{\mathbf{C}_{\alpha}}$ with
$-X_{\mathbf{C}_{\alpha}}$ if necessary, we have also
$|\nu_{i}-X_{\mathbf{C}_{\alpha}}(\cdot-y_{\alpha})|\leq c(\Lambda)\gamma$ on
$S_{i}\cap A_{2r_{\alpha},r_{\alpha}/8}(y_{\alpha})$, for each $\alpha$.

Consider the map $G_{i}:S_{i}\cap
B_{1+\sigma}\times(-2\epsilon_{S},2\epsilon_{S})\to\mathbb{R}^{8}$ by
$G_{i}(x^{\prime},t)=x^{\prime}+t\nu_{i}(x^{\prime})$. By definition of the
$\epsilon_{S}$, $G_{i}$ is a smooth diffeomorphism onto its image $O_{i}$, and
the images $O_{i}$ are disjoint. Provided $\gamma(\Lambda,\sigma)$ is
sufficiently small, we have $O_{i}\supset B_{2\epsilon_{S}}(S_{i})\cap
O^{\prime}$.

Define the smooth vector field $X_{i}=DG_{i}(\partial_{t})$ on $O_{i}$, so
that $t\mapsto G_{i}(x^{\prime},t)$ is the flow of $X_{i}$. Let $\xi(t)$ be a
smooth, increasing function satisfying

$\xi|_{(-\infty,1-2\sigma]}\equiv 0,\quad\xi|_{[1-\sigma,\infty)}\equiv
1,\quad|\xi^{\prime}|\leq 10/\sigma,$

and then define $Y_{i}$ on $B_{2\epsilon^{\prime}}(S_{i})\cap O^{\prime}$ by

$Y_{i}(x)=\left\\{\begin{array}[]{l
l}(1-\xi(|x|))X_{i}(x)+\xi(|x|)X_{\mathbf{C}}(x)&x\in A_{1,1-3\sigma}\\\
\xi(\frac{|x-y_{\alpha}|}{2r_{\alpha}})X_{i}(x)+(1-\xi(\frac{|x-y_{\alpha}|}{2r_{\alpha}}))X_{\mathbf{C}_{\alpha}}(x-y_{\alpha})&x\in
A_{2r_{\alpha},r_{\alpha}/5}(y_{\alpha})\\\
X_{i}(x)&otherwise\end{array}\right.$

For $Z_{i}=X_{i},Y_{i}$, we have

(81) $\sup_{B_{2\epsilon^{\prime}}(S_{i})\cap
A_{1,1-3\sigma}}|Z_{i}-X_{\mathbf{C}}|+|DZ_{i}-DX_{\mathbf{C}}|\leq c\gamma,$

and for each $\alpha$,

(82) $\sup_{B_{2\epsilon^{\prime}}(S_{i})\cap
A_{2r_{\alpha},r_{\alpha}/5}(y_{\alpha})}|Z_{i}-X_{\mathbf{C}_{\alpha}}(\cdot-
y_{\alpha})|+r_{\alpha}|DZ_{i}-DX_{\mathbf{C}_{\alpha}}(\cdot-y_{\alpha})|\leq
c\gamma,$

where $c=c(\Lambda,\sigma)$. And of course $X_{i}=Y_{i}$ in
$B_{1-3\sigma}\setminus\cup_{\alpha}B_{r_{\alpha}}(y_{\alpha})$. From the
above and (77), provided $\gamma(\Lambda,\sigma)$ is sufficiently small, the
flow $F_{X_{i}}(x^{\prime},t)$ is defined for all $|t|<\epsilon^{\prime}$,
$x^{\prime}\in S_{i}\cap O^{\prime}$.

Let $H_{i}:S_{i}\cap
O^{\prime}\times(-\epsilon^{\prime},\epsilon^{\prime})\to\mathbb{R}^{8}$ be
the smooth map given by $H_{i}(x^{\prime},t)=F_{X_{i}}(x^{\prime},t)$.
Recalling our restriction
$\epsilon^{\prime}\leq\epsilon_{0}\min_{\alpha}r_{\alpha}/16$, (81), (82),
(77) imply

(83) $\displaystyle|H_{i}-G_{i}|\leq
c(\Lambda,\sigma)\gamma|t|,\quad|DH_{i}-DG_{i}|\leq c(\Lambda,\sigma)\gamma$
(84) $\displaystyle 0.98|t|\leq d(G_{i}(x^{\prime},t),S_{i})\leq|t|$ (85)
$\displaystyle B_{\epsilon^{\prime}/2}(S_{i})\cap O^{\prime}\subset
H_{i}(S_{i}\cap O\times(-\epsilon^{\prime},\epsilon^{\prime}))\subset
B_{2\epsilon^{\prime}}(S_{i})\cap O^{\prime}.$

In particular, after identitying $S_{i}\cap O^{\prime}\times\mathbb{R}$ with
$T^{\perp}S_{i}$ via $\nu_{i}$, we have
$|DH_{i}-Id|\leq\epsilon^{\prime\prime}+c(\Lambda,\sigma)\gamma$. Ensuring
$\gamma(\Lambda,\sigma)$ and $\epsilon^{\prime\prime}(\Lambda)$ are
sufficiently small, we get that $H_{i}$ is a smooth diffeomorphism onto its
image.

By considering the map $H_{i}^{-1}\circ G_{i}$, ensuring
$\beta(\epsilon,\epsilon^{\prime\prime})$, $\gamma(\Lambda,\sigma)$ are
sufficiently small, and recalling the definition of smooth region, for each
$p=1,2,i=1,\ldots,k,j=1,\ldots,m_{k}$, we can find $C^{2}$ functions
$u_{pij}:S_{i}\to(-\epsilon^{\prime},\epsilon^{\prime})$ satisfying

(86) $\displaystyle M_{p}\cap
O^{\prime}=\cup_{i=1}^{k}\cup_{j=1}^{m_{i}}\\{H_{i}(x^{\prime},u_{pij}(x^{\prime})):x^{\prime}\in
S_{i}\cap O^{\prime}\\},$ (87)
$\displaystyle|u_{pij}|\leq(1+c\gamma)\beta\epsilon_{S}\leq\epsilon^{\prime}/10,\quad|u_{pij}|_{C^{1}}\leq
c(\epsilon^{\prime\prime}+\gamma+\beta\epsilon_{S})\leq 1,$

for $c=c(\Lambda,\sigma)$. Since each $M_{p}$ is multiplicity-one, by the
maximum principle we can assume that

(88) $\displaystyle u_{pi1}<u_{pi2}<\ldots<u_{pim_{k}}\quad\forall
p=1,2;i=1,\ldots,k.$

For each $i=1,\ldots,k$, define the $C^{2}$ diffeomorphism $g_{i}:S_{i}\cap
O\times(-\epsilon^{\prime},\epsilon^{\prime})\to S_{i}\cap
O\times(-\epsilon,\epsilon)$ by

$g_{i}(x^{\prime},t)=g_{i}(x^{\prime},f_{\\{u_{1ij}(x^{\prime})\\}_{j},\\{u_{2ij}(x^{\prime})\\}_{j}}(t)),$

where $f_{\\{a_{j}\\}_{j},\\{b_{j}\\}_{j}}(t)$ is the function from Lemma 12.1
with $m_{i}$ in place of $k$, and $\epsilon^{\prime}/10$ in place of
$\epsilon$. Then $g_{i}$ is the identity for $|t|>\epsilon/5$ and
$g_{i}(x^{\prime},u_{1ij}(x^{\prime}))=(x^{\prime},u_{2ij}(x^{\prime}))$ for
every $j=1,\ldots,m_{i}$. If we let

$D_{r}=\min\\{u_{1ij}(x^{\prime})-u_{1ij^{\prime}}(x^{\prime}):j<j^{\prime},x^{\prime}\in
S_{i}\cap B_{r}\cap O\\},$

then $D_{r}>0$ for all $r<1$ (by (88) and our definition of $O^{\prime}$), and
it follows by Lemma 12.1 and (87) that $|Dg_{i}|\leq
c(\Lambda,\sigma,D_{r})<\infty$ on $B_{r}$.

Define $\phi_{i}:O\to O$ by setting

$\phi_{i}(x)=H_{i}\circ g\circ H_{i}^{-1}\text{ if }x\in
B_{\epsilon^{\prime}/2}(S_{i})\cap O,\quad\phi_{i}(x)=x\text{ otherwise.}$

By (84), (85) and our construction of $g_{i}$, $\phi_{i}$ is a well-defined
$C^{2}$ diffeomorphism, satisfying

$\phi_{i}(\overline{M_{1}}\cap B_{\epsilon^{\prime}}(S_{i})\cap
O)=\overline{M_{2}}\cap B_{\epsilon^{\prime}}(S_{i})\cap
O,\quad|D\phi_{i}|\leq c(\Lambda,\sigma,M_{1}).$

Now set $\phi=\phi_{1}\circ\cdots\circ\phi_{k}$. We have $\phi(M_{1}\cap
O)=M_{2}\cap O$ by disjointness of the $O_{i}$, and $\phi|_{A_{1,1-\sigma}}$,
$\phi|_{A_{r_{\alpha},r_{\alpha}/4}(y_{\alpha})}$ are
$(\mathbf{C},\epsilon)$-, $(\mathbf{C}_{\alpha},\epsilon)$-maps by our
construction and our choice of $\gamma,\epsilon^{\prime}$. ∎

######  Lemma 8.5 (Cone region parameterization).

Take $\Lambda,\epsilon>0$, $\beta\geq 0$. Let
$\mathbf{C}\in\mathcal{C}_{\Lambda}$, $\rho\in[0,1)$, $m\in\mathbb{N}$. For
$i=1,2$, $j=1,\ldots,m$ let $u_{ij}:\mathbf{C}\cap A_{1,\rho}\to\mathbb{R}$ be
$C^{1}$ functions satisfying

$|x|^{-1}|u_{ij}|+|\nabla
u_{ij}|\leq\beta\leq\epsilon/4\leq\epsilon_{0}/8,\quad
u_{i1}<u_{i2}<\ldots<u_{im}\text{ for }i=1,2.$

Write $M_{i}=\cup_{j=1}^{m}G_{\mathbf{C}}(u_{ij}\nu_{\mathbf{C}})\cap
A_{1,\rho}$.

Then we can find a $C^{1}$ diffeomorphism $\phi:A_{1,\rho}\to A_{1,\rho}$
satisfying

(89)
$\displaystyle\phi(M_{1})=M_{2},\quad|\phi(x)|=|x|,\quad|D\phi|_{C^{0}(A_{r,\rho/r})}\leq
c(\Lambda,M_{1},r)\text{ for every $r<1$}$ (90) $\phi|_{A_{1,\rho}}$ is a
$(\mathbf{C},\epsilon)$-map.

If $m=1$, then in fact we have $|D\phi-Id|\leq
c(\Lambda)\frac{\beta}{\epsilon}$. If the $u_{ij}$ are $C^{k}$, then $\phi$ is
$C^{k}$ also.

######  Remark 8.6.

If $\rho=0$, the maximum principle [Ilm96] implies $m=1$, so provided
$\beta/\epsilon$ is sufficiently small (depending only on $\Lambda$), $\phi$
extends to a bi-Lipschitz map $B_{1}\to B_{1}$, satisfying

$(1-c(\Lambda)\beta/\epsilon)|x-y|\leq|\phi(x)-\phi(y)|\leq(1+c(\Lambda)\beta/\epsilon)|x-y|\quad\forall
x,y\in B_{1}.$

######  Remark 8.7.

It’s tempting to think the $\phi$ in Lemma 8.5 should also satisfy an estimate
like $|x||D^{2}\phi|\leq c(\Lambda,M_{1})$, provided the $u_{ij}$ satisfy a
similar estimate $|x||\nabla^{2}u_{ij}|\leq\beta$. This is certainly true if
$m=1$, but when $m\geq 2$ this seems to be false. The issue is that second
derivatives of the function $f_{\\{a_{j}\\}_{j},\\{b_{j}\\}_{j}}$ from Lemma
12.1 depend on the spacing of _both_ the $\\{a_{j}\\}_{j}$ and the
$\\{b_{j}\\}_{j}$ (as opposed to the first derivative which depends only on
the spacing of the $\\{a_{j}\\}_{j}$).

###### Proof.

Let $f_{\\{a_{j}\\}_{j},\\{b_{j}\\}_{j}}(t)$ be the function from Lemma 12.1,
with $m$ in place of $k$, $\epsilon/4$ in place of $\epsilon$, and let
$U=\\{(x,t)\in\mathbf{C}\cap
A_{1,\rho}\times\mathbb{R}:|t|<2\epsilon_{0}|x|\\}$. If $m\geq 2$, or
$\beta\geq 10^{-3}\epsilon$, define $g:U\to U$ by

$g(x,t)=(x,|x|f_{\\{|x|^{-1}u_{1j}(x)\\}_{j},\\{|x|^{-1}u_{2j}(x)\\}_{j}}(|x|^{-1}t)).$

If $m=1$ and $\beta\leq 10^{-3}\epsilon$, then take $\eta(t)$ to be the
function as in the proof of Lemma 12.1, and then instead set

$g(x,t)=(x,t+(u_{21}(x)-u_{11}(x))\eta(4\epsilon^{-1}|x|^{-1}(t-u_{11}(x)))).$

Then by Lemma 12.1 $g$ is a $C^{1}$ diffeomorphism $U\to U$, satisfying

$\displaystyle g(x,t)=(x,t)\text{ for
}|t|\geq\frac{\epsilon}{2}|x|,\quad|Dg|_{C^{0}(U\cap A_{r,\rho/r})}\leq
c(\Lambda,M_{1},r)\text{ for all $r<1$},$ $\displaystyle
g(x,u_{1j}(x))=(x,u_{2j}(x))\text{ for each }j=1,\ldots,m.$

If $m=1$, then $|Dg-Id|\leq c(\Lambda)\beta/\epsilon$ on all of $U$.

Now define $\phi$ by

$\phi(x)=\left\\{\begin{array}[]{l l}G_{\mathbf{C}}\circ g\circ
G_{\mathbf{C}}^{-1}&x\in A_{1,\rho}\cap G_{\mathbf{C}}(U)\\\ x&x\in
A_{1,\rho}\setminus G_{\mathbf{C}}(U)\end{array}\right..\qed$

######  Remark 8.8.

If $\rho=0$ and $\mathbf{C}$ is “integrable through rotations” (e.g. if
$\mathcal{C}(\mathbf{C})$ consists of rotations of $\mathbf{C}$), then
provided $\beta(\mathbf{C},\epsilon)$, $\epsilon(\Lambda)$ are sufficiently
small, instead of (89), (90) we can arrange $\phi$ to instead satisfy:

(91) $\displaystyle\text{$\phi$ is a $C^{2}$ diffeomorphism $A_{1,0}\to
A_{1,0}$},\quad\phi(M_{1})=M_{2},\quad|\phi(x)|=|x|,$ (92)
$\displaystyle\phi\text{ extends in a $C^{1,\alpha}$ fashion to $B_{1}$, for
some $\alpha(\mathbf{C})>0$},$ (93) $\displaystyle\text{$\phi|_{A_{1,1/8}}$ is
a $(\mathbf{C},\epsilon)$-map}.$

By using this $\phi$ in the proof of Theorem 8.1 instead of the $\phi$
generated by Lemma 8.5, one can get the map in Theorem 8.1 to be globally
$C^{1,\alpha}$.

We outline how to do this. [AA81] (see also [Sim93, Lemma 1]) implies that for
such $\mathbf{C}$ there are constants
$\alpha(\mathbf{C}),\delta(\mathbf{C})\in(0,1)$ so that if
$M\in\mathcal{M}_{7}(B_{1},g)$, $M\cap(A_{1,0},g)$ is a
$(\mathbf{C},1,\delta)$-strong-cone region, and
$|g-g_{\mathrm{eucl}}|_{C^{3}(B_{1})}\leq\delta$, then there is a
$\mathbf{C}^{\prime}\in\mathcal{C}(\mathbf{C})$ being a rotation of
$\mathbf{C}$ such that

$M\cap A_{1,0}=G_{\mathbf{C}^{\prime}}(u)\cap A_{1,0},\quad|x|^{-1}|u|+|\nabla
u|+|x||\nabla^{2}u|\leq c(\mathbf{C})\delta|x|^{\alpha},$

for some $C^{2}$ function $u:\mathbf{C}^{\prime}\cap
A_{1,0}\to\mathbf{C}^{\prime\perp}$.

Now take $M_{1},M_{2}$ satisfying the hypotheses of Lemma 8.5 with this
$\mathbf{C}$, and $\beta(\mathbf{C},\epsilon)$, $\epsilon(\Lambda)$ to be
chosen. Combining the previous paragraph with Lemma 8.5, provided
$\beta(\mathbf{C},\epsilon)$ is small we deduce there are
$\mathbf{C}_{1},\mathbf{C}_{2}$ rotations of $\mathbf{C}$ and $C^{2}$
diffeomorphisms $\phi_{1},\phi_{2}:A_{1,0}\to A_{1,0}$, so that
$\phi_{i}(\mathbf{C}_{i}\cap A_{1,0})=\mathrm{spt}M_{i}\cap A_{1,0}$,
$|D\phi_{i}|_{x}-Id|\leq c(\mathbf{C})\beta|x|^{\alpha}$, and
$\phi_{i}|_{A_{1,0}}$ is a $(\mathbf{C}_{i},\epsilon)$-map. In particular each
$\phi_{i}$ extends to a $C^{1,\alpha}$ map on $B_{1}$ with
$D\phi_{i}|_{0}=Id$. Let $P$ be any rotation taking $\mathbf{C}_{1}$ to
$\mathbf{C}_{2}$, with $|P-Id|\leq c(\mathbf{C})\beta$.

On the other hand, we can also apply Lemma 8.5 directly to obtain a $C^{2}$
diffeomorphism $\phi$ satisfying (89), (90). Since
$d_{H}(\mathbf{C}_{i}\cap\partial B_{1},\mathbf{C}\cap\partial B_{1})\leq
c(\mathbf{C})\beta$, using Theorem 5.1 and Lemma 8.9, ensuring
$\epsilon(\Lambda)$ and $\beta(\Lambda,\epsilon)$ are sufficiently small, we
can obtain a $C^{2}$ diffeomorphism $\psi:A_{1,0}\to A_{1,0}$ satisfying

$\psi|_{A_{1,1/8}}=\phi,\quad\psi_{A_{1/16,0}}=\phi_{2}\circ
P\circ\phi_{1}^{-1},\quad|\psi(x)|=|x|,\quad|D\psi|\leq c(\mathbf{C}).$

This $\psi$ will satisfy (91), (92), (93).

Unfortunately, integrability is in general an open condition, so it’s not
clear if one can take $\beta$ to depend only on $\Lambda$, rather than the
specific integrable cone $\mathbf{C}$. Of course, the only _known_ stable
$7$-dimensional hypercones are the quadratic cones, which are all integrable
through rotations, and one could obviously arrange $\beta$ so that the
previous argument works with these cones. In fact I believe there are no known
minimal hypercones which are not integrable through rotations. At any rate it
would be interesting to understand this better.

######  Lemma 8.9 (Gluing Lemma).

Given $\Lambda,\epsilon>0$, $\sigma\in(0,10^{-2}]$, there are constants
$\delta_{1}(\Lambda,\sigma)$, $\delta^{\prime}_{1}(\Lambda,\epsilon,\sigma)$,
$\epsilon_{1}(\Lambda)\leq\epsilon_{0}/4$ so that provided
$\epsilon\leq\epsilon_{1}$ the following holds. Let
$\mathbf{C}\in\mathcal{C}_{\Lambda}$, $m\in\mathbb{N}$. For
$i=1,2,j=1,\ldots,m$, take $C^{1}$ functions $u_{ij}:\mathbf{C}\cap
A_{1+\sigma,1-4\sigma}\to\mathbb{R}$ satisfying

$|u_{ij}|_{C^{1}}\leq\epsilon/10,\quad u_{i1}<u_{i2}<\ldots<u_{im},$

and then set $M_{i}=\cup_{j=1}^{m}G_{\mathbf{C}}(u_{ij}\nu_{\mathbf{C}})\cap
A_{1+\sigma,1-4\sigma}$.

Let $O_{1},O_{2}\supset A_{1+\sigma,1-4\sigma}$ be open sets in
$\mathbb{R}^{8}$. For $l=1,2$, let $T_{l}:O_{l}\to\mathbb{R}^{8}$,
$\phi_{l}:O_{l}\to O_{l}$ be $C^{1}$ diffeomorphisms such that

$\displaystyle|T_{l}-id|_{C^{1}(O_{l})}\leq\delta^{\prime}_{1},\quad
T_{l}(O_{l})\supset A_{1+\sigma,1-4\sigma},$ $\displaystyle
T_{l}(\phi_{l}(M_{1}\cap O_{l}))=M_{2}\cap T_{l}(O_{l}),\quad\phi_{l}=id\text{
outside }B_{\epsilon/10}(\mathbf{C}).$

Assume additionally there are smooth vector fields $X_{l}$ on $O_{l}\cap
B_{2\epsilon}(\mathbf{C})$ satisfying

$|X_{l}-X_{\mathbf{C}}|_{C^{1}(B_{2\epsilon}(\mathbf{C})\cap
O_{l})}\leq\delta_{1},$

so that each $\phi_{l}$ preserves the flowlines of $X_{l}$.

Define

$\displaystyle O=(O_{1}\setminus B_{1-2\sigma})\cup(O_{2}\cap B_{1-\sigma}),$
$\displaystyle O^{\prime}=(T_{1}(O_{1})\setminus
B_{1-2\sigma})\cup(T_{2}(O_{2})\cap B_{1-\sigma}).$

Then we can find a $C^{1}$ diffeomorphism $h:O\to O^{\prime}$ with the
properties that:

$\displaystyle h|_{O\setminus B_{1-0.5\sigma}}=T_{1}\phi_{1},\quad h|_{O\cap
B_{1-2.5\sigma}}=T_{2}\phi_{2},$ $\displaystyle h(M_{1}\cap O)=M_{2}\cap
O^{\prime},\quad|Dh|\leq
c(\Lambda,\sigma,M_{1},|D\phi_{1}|_{C^{1}(O_{1})},|D\phi_{2}|_{C^{1}(O_{2})}).$

If the $u_{ij}$, $T_{l}$, $\phi_{l}$ are $C^{k}$, then $h$ is $C^{k}$ also.

######  Remark 8.10.

Of course the same Lemma holds true if $X_{\mathbf{C}}$ is replaced with
$-X_{\mathbf{C}}$.

###### Proof.

Take $\epsilon\leq\epsilon_{0}/4$. For ease of notation write
$U=(\mathbf{C}\cap A_{1,1-3\sigma})\times(-\epsilon,\epsilon)$. Fix $\xi(t)$
to be a smooth, increasing function satisfying

$\xi|_{(-\infty,1-2\sigma]}=0,\quad\xi|_{[1-\sigma,\infty)}=1,\quad\xi^{\prime}\leq
10/\sigma.$

Define $T:A_{1+\sigma,1-4\sigma}\to\mathbb{R}^{8}$ by
$T(x)=\xi(|x|)T_{1}(x)+(1-\xi(|x|))T_{2}(x)$. Then for
$\delta^{\prime}(\sigma)$ sufficiently small, $T$ is a $C^{1}$ diffeomorphism
onto its image satisfying

$|T-id|\leq\delta^{\prime},\quad|DT-Id|\leq c(\sigma)\delta^{\prime}.$

Observe that from our hypotheses on $\phi_{l}$ and our construction of $T$, we
have

(94) $\displaystyle T_{1}\phi_{1}=T\text{ on }A_{1,1-\sigma}\setminus
B_{\epsilon/10}(\mathbf{C}),\quad T_{2}\phi_{2}=T\text{ on
}A_{1-2\sigma,1-3\sigma}\setminus B_{\epsilon/10}(\mathbf{C}).$

Define on $B_{2\epsilon}(\mathbf{C})\cap A_{1+\sigma,1-4\sigma}$ the smooth
vector field

$X(x)=\xi(|x|)X_{1}(x)+(1-\xi(|x|))X_{2}(x).$

Then we have

$|X-X_{\mathbf{C}}|_{C^{1}(B_{2\epsilon}(\mathbf{C})\cap
A_{1+\sigma,1-4\sigma})}\leq c(\sigma)\delta.$

By the above and (77), provided $\delta(\sigma)$ is sufficiently small, the
flow $F_{X}(x^{\prime},t)$ of $X$ exists for all $|t|<\epsilon$ and
$x^{\prime}\in\mathbf{C}\cap A_{1,1-3\sigma}$.

Define $G:U\to\mathbb{R}^{8}$ by $G(x^{\prime},t)=F_{X}(x^{\prime},t)$. Then,
ensuring $\epsilon(\Lambda)$ and $\delta(\Lambda,\sigma)$ are small, $G$ is a
smooth diffeomorphism onto its image satisfying:

(95) $\displaystyle|G-G_{\mathbf{C}}|\leq c\delta|t|,\quad|DG-
DG_{\mathbf{C}}|+|D(G^{-1}\circ G_{\mathbf{C}})-Id|\leq c\delta,$ (96)
$\displaystyle(1-c\delta)|x^{\prime}|\leq|G(x^{\prime},t)|\leq(1+c\delta)|x^{\prime}|$
(97) $\displaystyle 0.98|t|\leq d(G(x^{\prime},t),\mathbf{C})\leq 1.02|t|,$

for $c=c(\Lambda,\sigma)$, and

(98) $\displaystyle B_{0.9\epsilon}(\mathbf{C})\cap
A_{1-0.1\sigma,1-2.9\sigma}\subset G(U)\subset B_{1.1\epsilon}(\mathbf{C})\cap
A_{1+0.1\sigma,1-3.1\sigma},$

Similarly, ensuring $\delta^{\prime}(\epsilon,\sigma)$ is small, then the
diffeomorphism $TG\equiv T\circ G$ satisfies (with $c=c(\Lambda,\sigma)$)

(99) $\displaystyle|TG-G_{\mathbf{C}}|\leq
c(\delta|t|+\delta^{\prime}),\quad|D(TG)-G_{\mathbf{C}}|+|D(G^{-1}T^{-1}G_{\mathbf{C}})-Id|\leq
c(\delta+\delta^{\prime})$ (100) $\displaystyle
B_{0.8\epsilon}(\mathbf{C})\cap
A_{1-0.2\sigma,1-2.8\sigma}\subset(TG)(U)\subset
B_{1.2\epsilon}(\mathbf{C})\cap A_{1+0.2\sigma,1-3.2\sigma}.$

By considering the maps $G^{-1}G_{\mathbf{C}}$ and
$G^{-1}T^{-1}G_{\mathbf{C}}$, ensuring
$\delta^{\prime}(\epsilon,\Lambda,\sigma)$, $\delta(\Lambda,\sigma)$ are
small, then (95)-(100) imply we can find $C^{1}$ functions
$u^{G}_{j},u^{T}_{j}:U\to\mathbb{R}$ ($j=1,\ldots,m$) satisfying

$\displaystyle M_{1}\cap
G(U)=\cup_{j=1}^{m}\\{G(x^{\prime},u^{G}_{j}(x^{\prime})):x^{\prime}\in U\\},$
(101)
$\displaystyle|u^{G}_{j}|\leq(1+c\delta)\epsilon/10\leq\epsilon/9,\quad|u^{G}_{j}|_{C^{1}(U)}\leq
c(\beta+\delta)\leq 1,$ $\displaystyle u^{G}_{1}<u^{G}_{2}<\ldots<u^{G}_{m},$

and

$\displaystyle M_{2}\cap
TG(U)=\cup_{j=1}^{m}\\{TG(x^{\prime},u^{T}_{j}(x^{\prime})):x^{\prime}\in
U\\},$ (102)
$\displaystyle|u^{T}_{j}|\leq(1+c\delta)\epsilon/10+c\delta^{\prime}\leq\epsilon/9,\quad|u^{T}_{j}|_{C^{1}(U)}\leq
c(\beta+\delta+\delta^{\prime})\leq 1,$ $\displaystyle
u^{T}_{1}<u^{T}_{2}<\ldots<u^{T}_{m},$

where $c=c(\Lambda,\sigma)$.

By (94), (96) (taking $\delta(\Lambda,\sigma)$ sufficiently small), (97), and
our hypotheses on $T_{l}$, $\phi_{l}$, we can define $C^{1}$ functions
$t_{1}:\mathbf{C}\cap
A_{1,1-0.9\sigma}\times(-\epsilon,\epsilon)\to\mathbb{R}$,
$t_{2}:\mathbf{C}\cap A_{1-2.1\sigma,1-3\sigma}\to\mathbb{R}$ by

$(G^{-1}\circ T^{-1}\circ T_{l}\circ\phi_{l}\circ
G)(x^{\prime},t)=(x^{\prime},t_{l}(x^{\prime},t)).$

The $t_{l}$ have the properties that: for each $x^{\prime}$,
$t_{l}(x^{\prime},\cdot)$ is a diffeomorphism of $(-\epsilon,\epsilon)$
coinciding with $id$ for $|t|>\epsilon/9$;
$t_{l}(x^{\prime},u_{j}^{G}(x^{\prime}))=u_{j}^{T}(x^{\prime})$; and

(103) $\displaystyle|\nabla t_{l}|+|\partial_{t}t_{l}|\leq
c(\Lambda,\sigma,|\phi_{l}|_{C^{1}(O_{l})}).$

Let $\eta_{1},\eta_{2}:\mathbb{R}\to\mathbb{R}$ be a smooth functions
satisfying

$\displaystyle\eta_{1}|_{(-\infty,1-2.3\sigma]}\equiv
1,\quad\eta_{1}|_{[1-2.2\sigma,\infty)}\equiv 0,\quad
0\leq-\eta_{1}^{\prime}\leq 100/\sigma$
$\displaystyle\eta_{2}|_{(-\infty,1-0.8\sigma]}\equiv
1,\quad\eta_{2}|_{[1-0.7\sigma,\infty)}\equiv 0,\quad
0\leq-\eta_{2}^{\prime}\leq 100/\sigma.$

Define $f:U\to\mathbb{R}$ by

$f(x^{\prime},t)=f_{\\{u_{j}^{G}(x^{\prime})\\}_{j},\\{u_{j}^{T}(x^{\prime})\\}_{j}}(t)$

where $f_{\\{a_{j}\\}_{j},\\{b_{j}\\}_{j}}(t)$ is the function from Lemma 12.1
with $\epsilon/9$ in place of $\epsilon$, and $m$ in place of $k$. Define
$s:U\to\mathbb{R}$ by

$\displaystyle s(x^{\prime},t)=\left\\{\begin{array}[]{l
l}\eta_{1}(|x^{\prime}|)t_{2}(x^{\prime},t)+(1-\eta_{1}(|x^{\prime}|))f(x^{\prime},t)&1-3\sigma<|x^{\prime}|<1-2.2\sigma\\\
f(x^{\prime},t)&1-2.2\sigma\leq|x^{\prime}|<1-0.8\sigma\\\
\eta_{2}(|x^{\prime}|)f(x^{\prime},t)+(1-\eta_{2}(|x^{\prime}|))t_{1}(x^{\prime},t)&1-0.8\sigma\leq|x^{\prime}|<1\end{array}\right..$

Then $s$ is $C^{1}$ (or $C^{k}$ if the $\phi_{l}$, $T_{l}$, $u_{ij}$ are
$C^{k}$), and $s(x^{\prime},\cdot)$ is a diffeomorphism of
$(-\epsilon,\epsilon)$ for every $x^{\prime}$, and $s(x^{\prime},t)$ enjoys
the properties:

$\displaystyle s=t_{1}\text{ if }|x^{\prime}|>1-0.7\sigma,\quad s=t_{2}\text{
if }|x^{\prime}|<1-2.3\sigma,$ $\displaystyle s(x^{\prime},t)=t\text{ if
}|t|\geq\epsilon/4$ $\displaystyle
s(x^{\prime},u_{j}^{G}(x^{\prime}))=u_{j}^{T}(x^{\prime})\text{ for every
}j=1,\ldots,m.$

If we let

$D=\inf\\{u_{j}^{G}(x^{\prime})-u_{j^{\prime}}^{G}(x^{\prime}):j<j^{\prime},x^{\prime}\in\mathbf{C}\cap
A_{1-0.5\sigma,1-2.5\sigma}\\}>0,$

then by (101), (102), (103) and Lemma 12.1 we get

$|\nabla s|+|\partial_{t}s|\leq
c(\Lambda,\sigma,D,|\phi_{1}|_{C^{1}(O_{1})},|\phi_{2}|_{C^{1}(O_{2})})<\infty.$

Let us define $g:U\to U$ by $g(x^{\prime},t)=(x^{\prime},s(x^{\prime},t))$.
Trivially $g$ is a $C^{1}$ (or $C^{k}$) diffeomorphism of $U$. We then define
$h:O\to O^{\prime}$ by

$\displaystyle h(x)=\left\\{\begin{array}[]{l l}(T\circ G\circ g\circ
G^{-1})(x)&x\in B_{0.9\epsilon}(\mathbf{C})\cap A_{1-0.1\sigma,1-2.9\sigma}\\\
T(x)&x\in A_{1-0.1\sigma,1-2.9\sigma}\setminus B_{0.9\epsilon}(\mathbf{C})\\\
T_{1}\phi_{1}(x)&x\in O\setminus B_{1-0.2\sigma}\\\ T_{2}\phi_{2}(x)&x\in
O\cap B_{1-2.8\sigma}\end{array}\right..$

It follows by (95)-(98) and our definition of $g$ that $\phi$ is well-defined,
and satisfies the requirements of the Lemma. ∎

We are now set up to prove Theorem 8.1.

###### Proof of Theorem 8.1.

We prove Theorem 8.1 by induction on $N$ and by contradiction. Suppose either
$N=1$, or by inductive hypothesis the Lemma holds with $N-1$ in place of $N$.
Suppose, towards a contradiction, the Lemma fails, for $\beta,\gamma,\epsilon$
for the moment arbitrarily fixed, but to be chosen later (we note that there
is no loss in assuming $\epsilon(\Lambda,\sigma)$ is as small as we like).
Then there is a finite collection $\mathcal{S}$ of
$(\Lambda,\sigma,\gamma)$-smooth models, a set $\mathcal{G}$ of $C^{3}$
metrics on $B_{1}$, sequences $g_{i}\in\mathcal{G}$,
$M_{i}\in\mathcal{M}_{7}(B_{1},g_{i})$,
$\mathbf{C}_{i}\in\mathcal{C}_{\Lambda}$, and an $r<1$, so that each
$M_{i}\cap B_{1}$ admits a $(\Lambda,\beta,\mathcal{S},N)$-strong-cone
decomposition, but for every $j>i$ there is no local bi-Lipschitz map
$\phi_{ij}:B_{1}\to B_{1}$ satisfying $Lip(\phi_{ij}|_{B_{r}})\leq j$, and
making Theorem 8.1(1),(2),(4),(5) true with $\phi_{ij}$, $M_{i}$,
$\mathbf{C}_{i}$, $M_{j}$ in place of $\phi$, $M_{v}$, $\mathbf{C}_{v}$, $M$
(respectively). Note that if we replace $i$ with any subsequence of $i$, then
this new sequence satisfies the same contradiction hypotheses.

By hypothesis, for every $i$ there is a collection of at most $N$ smooth
regions and strong-cone regions which fit together as per Definition 7.0.4.
After passing to a subsequence (and apply Theorem 5.1), we can assume that for
all $i$ we have either: $M_{i}\cap(B_{1},g_{i})$ is a $(S,\beta)$-smooth
region for some fixed smooth model
$(S,\mathbf{C},\\{(\mathbf{C}_{\alpha},B_{r_{\alpha}}(y_{\alpha}))\\}_{\alpha})\in\mathcal{S}$;
or $M_{i}\cap(A_{1,\rho_{i}},g_{i})$ is a $(\mathbf{C},m,2\beta)$-strong-cone
region for some $\mathbf{C}\in\mathcal{C}_{\Lambda}$, $m\in\mathbb{N}$,
$\rho_{i}\leq 1/2$.

We treat each case separately, but observe first that by definition of
cone-/smooth-region, we have for each $i$:

$\displaystyle d_{H}(M_{i}\cap A_{1,1-\sigma},\mathbf{C}\cap
A_{1,1-\sigma})\leq c(\Lambda)(\beta+\gamma)\leq\epsilon,$
$\displaystyle\mathrm{sing}M_{i}\subset B_{1-\sigma},$

provided we take
$\beta(\Lambda,\sigma,\epsilon),\gamma(\Lambda,\sigma,\epsilon)$ sufficiently
small. So Theorem 8.1(5) holds for $M_{i}$, $\mathbf{C}$ in place of $M$,
$\mathbf{C}_{v}$ (resp.) for all $i$.

Case 1: Every $M_{i}\cap(B_{1},g_{i})$ is a $(S,\beta)$-smooth region. Let
$O=B_{1}\setminus\cup_{\alpha}B_{r_{\alpha}/4}(y_{\alpha})$. Taking
$\gamma(\Lambda,\sigma,\epsilon)$, $\beta(\Lambda,\epsilon)$ sufficiently
small, we can apply Lemma 8.4 to obtain a $C^{2}$ diffeomorphism
$\psi_{i}:O\to O$ satisfying

$\displaystyle\psi_{i}(\overline{M_{1}}\cap O)=\overline{M_{i}}\cap
O,\quad|D\psi_{i}|_{C^{0}(O\cap B_{r})}\leq c(\Lambda,\sigma,M_{1},r)\text{
for any }r<1$ $\displaystyle\psi_{i}|_{A_{1,1-\sigma}}\text{ is a
$(\mathbf{C},\epsilon)$-map},$
$\displaystyle\psi_{i}|_{A_{r_{\alpha},r_{\alpha}/4}(y_{\alpha})}\text{ is a
$(\mathbf{C}_{\alpha},\epsilon)$-map, for each $\alpha$.}$

If $\\{\alpha\\}=\emptyset$, then $O=B_{1}$, and we deduce a contradiction.
This proves Case 1 when $N=1$.

By our inductive hypothesis, passing to a further subsequence as necessary, we
can find for every $\alpha$ a $g_{\alpha}\in\mathcal{G}$,
$\hat{M}_{\alpha}\in\mathcal{M}_{7}(B_{1},g_{\alpha})$,
$\hat{\mathbf{C}}_{\alpha}\in\mathcal{C}_{\Lambda}$, $x_{\alpha i}\in B_{\beta
r_{\alpha}}(y_{\alpha})$, $r_{\alpha
i}\in(\frac{1}{2}r_{\alpha},(1+\beta)r_{\alpha})$, and local bi-Lipschitz maps
$\hat{\phi}_{\alpha i}:B_{1}\to B_{1}$, so that Theorem 8.1(1)-(5) hold with
$\hat{\phi}_{\alpha i}$, $\hat{M}_{\alpha}$, $\hat{\mathbf{C}}_{\alpha}$,
$\eta_{x_{\alpha i},r_{\alpha i}}(M_{i})$, $C_{\mathcal{S},N-1,\mathcal{G}}$
in place of $\phi$, $M_{v}$, $\mathbf{C}_{v}$, $M$,
$C_{\mathcal{S},N,\mathcal{G}}$ (respectively). We can moreover assume that
the $\\{r_{\alpha i}\\}_{i}$ converge, and hence assume
$1-\beta\leq\frac{r_{\alpha i}}{r_{\alpha 1}}\leq 1+\beta$ for for all $i$.

By virtue of $M_{i}\cap(B_{1},g_{i})$ being a smooth region, if we ensure
$\gamma(\Lambda)$, $\beta(\Lambda)$ are sufficiently small and recall that
$\epsilon_{S}\leq\min_{\alpha}r_{\alpha}$, then we can write

(104) $M_{i}\cap
A_{2r_{\alpha},r_{\alpha}/4}(y_{\alpha})=\cup_{j=1}^{m}G_{y_{\alpha}+\mathbf{C}_{\alpha}}(u_{\alpha
ij})\cap A_{2r_{\alpha},r_{\alpha}/8}(y_{\alpha})$

for $C^{2}$ functions $u_{\alpha ij}:(y_{\alpha}+\mathbf{C}_{\alpha})\cap
A_{2r_{\alpha},r_{\alpha}/8}(y_{\alpha})\to\mathbf{C}_{\alpha}^{\perp}$
satisfying

(105) $r_{\alpha}^{-1}|u_{\alpha ij}|+|\nabla u_{\alpha
ij}|+r_{\alpha}|\nabla^{2}u_{\alpha ij}|\leq
c(\Lambda)(\gamma+\beta)\leq\epsilon_{1}(\Lambda)/100$

where $\epsilon_{1}$ as in Lemma 8.9.

Ensure $\beta\leq 10^{-3}\sigma$, and let $\sigma^{\prime}=10^{-2}\sigma$,
$r_{\alpha}^{\prime}=\frac{1-2\sigma^{\prime}}{1+5\sigma^{\prime}}\min\\{r_{\alpha},r_{\alpha
1}\\}$. Then we have

$A_{\alpha}:=A_{(1+\sigma^{\prime})r_{\alpha}^{\prime},(1-4\sigma^{\prime})r_{\alpha}^{\prime}}(y_{\alpha})\subset
A_{r,(1-\sigma)r}(z),$

for any $z\in B_{\beta r_{\alpha}}(y_{\alpha})$ and any
$r\in((1-10\beta)r_{\alpha 1},(1+10\beta)r_{\alpha 1})$. Note in particular
this holds for $z=x_{\alpha i}$ and $r=r_{\alpha i}$.

From (104),(105), Theorem 8.1(5), and the bounds $|y_{\alpha}-x_{\alpha
i}|\leq\beta r_{\alpha}$, $\frac{1}{2}\leq\frac{r_{\alpha
i}}{r_{\alpha}}\leq\frac{3}{2}$, we get

$d_{H}(\mathbf{C}_{\alpha}\cap B_{1},\hat{\mathbf{C}}_{\alpha}\cap B_{1})\leq
c(\Lambda)(\gamma+\beta+\epsilon).$

Recalling that $\epsilon_{1}\leq\epsilon_{0}/4$ (for $\epsilon_{1}(\Lambda)$
as in Lemma 8.9, $\epsilon_{0}(\Lambda)$ as in the earlier discussion in this
Section), and ensuring $\gamma,\beta,\epsilon$ are sufficiently small
(depending only on $\Lambda,\sigma$), we deduce that both
$X_{\hat{\mathbf{C}}_{\alpha}}(x-x_{\alpha 1})$ and
$X_{\mathbf{C}_{\alpha}}(x-y_{\alpha})$ are defined for $x\in
B_{2\epsilon_{1}r^{\prime}_{\alpha}}(y_{\alpha}+\mathbf{C}_{\alpha})\cap
A_{\alpha}$ and satisfy

$\displaystyle|X_{\hat{\mathbf{C}}_{\alpha}}(x-x_{\alpha
1})-X_{\mathbf{C}_{\alpha}}(x-y_{\alpha})|$
$\displaystyle\quad+r_{\alpha}^{\prime}|DX_{\hat{\mathbf{C}}_{\alpha}}(x-x_{\alpha
1})-DX_{\mathbf{C}_{\alpha}}(x-y_{\alpha})|\leq\delta_{1}(\Lambda,\sigma^{\prime}).$

Here $\delta_{1}(\Lambda,\sigma^{\prime})$ are the constant from Lemma 8.9.
Finally, if we set $T_{\alpha i}=\eta_{x_{\alpha i},r_{\alpha
i}}^{-1}\circ\eta_{x_{\alpha 1},r_{\alpha 1}}$, then provided we take
$\beta(\Lambda,\sigma)$ small we get

$r_{\alpha}^{\prime-1}|T_{\alpha i}-id|+|DT_{\alpha
i}-Id|\leq\delta_{1}^{\prime}(\Lambda,\epsilon_{1}(\Lambda),\sigma^{\prime})\text{
on }B_{2r^{\prime}_{\alpha}}(y_{\alpha}),$

where $\delta_{1}^{\prime}(\Lambda,\epsilon_{1}(\Lambda),\sigma^{\prime})$ as
in Lemma 8.9.

Now define the local bi-Lipschitz maps $\phi_{\alpha i}:B_{r_{\alpha
1}}(x_{\alpha 1})\to B_{r_{\alpha i}}(x_{\alpha i})$ by

$\phi_{\alpha i}(x)=(\eta_{x_{\alpha i},r_{\alpha
i}})^{-1}\circ\hat{\phi}_{\alpha i}\circ\hat{\phi}_{\alpha
1}^{-1}\circ\eta_{x_{\alpha 1},r_{\alpha i}}.$

The $\phi_{\alpha i}$ restrict to $C^{2}$ diffeomorphisms $A_{r_{\alpha
1},(1-\sigma)r_{\alpha 1}}(x_{\alpha 1})\to A_{r_{\alpha
i},(1-\sigma)r_{\alpha i}}(x_{\alpha i})$, and map $\mathrm{spt}M_{1}\cap
B_{r_{\alpha 1}}(x_{\alpha 1})$ to $\mathrm{spt}M_{i}\cap B_{r_{\alpha
i}}(x_{\alpha i})$, and admit the bound $\mathrm{Lip}(\phi_{\alpha
i}|_{B_{(1+\sigma^{\prime})r_{\alpha}^{\prime}}(y_{\alpha})})\leq
C_{\mathcal{S},N-1,\mathcal{G}}(\frac{1+\sigma^{\prime}}{1+5\sigma^{\prime}})$.
Moreover, from Theorem 8.1(4) and our definition of $A_{\alpha}$ we have that

$\displaystyle T_{\alpha i}^{-1}\phi_{\alpha i}|_{A_{\alpha}}\text{ preserves
the flow lines of $X_{\hat{\mathbf{C}}_{\alpha}}(\cdot-x_{\alpha 1})$},$

and

$\displaystyle T_{\alpha i}^{-1}\phi_{\alpha i}|_{A_{\alpha}}=id\text{ outside
}$ $\displaystyle B_{\epsilon r_{\alpha}}(x_{\alpha
1}+\hat{\mathbf{C}}_{\alpha})$ $\displaystyle\subset
B_{c(\Lambda)(\gamma+\beta+\epsilon)r_{\alpha}}(y_{\alpha}+\mathbf{C}_{\alpha})$
$\displaystyle\subset
B_{\epsilon_{1}(\Lambda)r_{\alpha}^{\prime}/10}(y_{\alpha}+\mathbf{C}_{\alpha})$

provided $\gamma,\beta,\epsilon$ are sufficiently small (depending only on
$\Lambda,\sigma$).

On the other hand, since $A_{\alpha}\subset
A_{r_{\alpha},r_{\alpha}/4}(y_{\alpha})$, by construction
$\psi_{i}|_{A_{\alpha}}$ preserves the flowlines of
$X_{\mathbf{C}_{\alpha}}(\cdot-y_{\alpha})$, and $\psi_{i}|_{A_{\alpha}}=id$
outside $B_{\epsilon r_{\alpha}}(y_{\alpha}+\mathbf{C}_{\alpha})\subset
B_{\epsilon_{1}(\Lambda)r_{\alpha}^{\prime}/10}(y_{\alpha}+\mathbf{C}_{\alpha})$.

The above discussion implies we can apply Lemma 8.9 to the maps $\psi_{i}$,
$\phi_{\alpha i}$ at scale $B_{r_{\alpha}^{\prime}}(y_{\alpha})$ (with
$\sigma^{\prime}$ in place of $\sigma$), to obtain local bi-Lipschitz maps
$\phi_{i}:B_{1}\to B_{1}$ satisfying:

$\displaystyle\phi_{i}|_{B_{1}\setminus\cup_{\alpha}B_{r_{\alpha}^{\prime}}(y_{\alpha})}=\psi_{i},\quad\phi_{i}|_{B_{(1-3\sigma^{\prime})r^{\prime}_{\alpha}}(y_{\alpha})}=\phi_{\alpha
i}$ $\displaystyle\phi_{i}(\overline{M_{1}}\cap B_{1})=\overline{M_{i}}\cap
B_{1},\quad\phi_{i}(\mathrm{sing}M_{1})=\mathrm{sing}M_{i}$
$\displaystyle\phi_{i}|_{B_{1}\setminus\mathrm{sing}M_{1}}\text{ is a $C^{2}$
diffeomorphism}$ $\displaystyle\quad\mathrm{Lip}(\phi_{i}|_{B_{r}})\leq
c(\Lambda,\sigma,M_{1},C_{\mathcal{S},N-1,\mathcal{G}},r)\text{ for }r<1.$

This is a contradiction, and therefore completes the proof of Case 1.

Case 2: Every $M_{i}\cap(A_{1,\rho_{i}},g_{i})$ is a
$(\mathbf{C},m,2\beta)$-strong-cone region. Passing to a subsequence, we can
assume that either $\rho_{i}=0$ for all $i$, or $\rho_{i}>0$ for all $i$. By
the maximum principle, we can write

(106) $\displaystyle M_{i}\cap
A_{1,\rho_{i}/8}=\cup_{j=1}^{m}G_{\mathbf{C}}(u_{ij}\nu_{\mathbf{C}})\cap
A_{1,\rho_{i}}$

for $C^{2}$ functions $u_{ij}:\mathbf{C}\cap A_{1,\rho_{i}/8}\to\mathbb{R}$
satisfying

(107) $\displaystyle|x|^{-1}|u_{ij}|+|\nabla u_{ij}|+|x||\nabla^{2}u_{ij}|\leq
2\beta,\quad u_{i1}<u_{i2}<\ldots<u_{im}.$

Let us ensure that
$2\beta\leq\min\\{\epsilon/c(\Lambda),\epsilon_{0}(\Lambda)/8\\}$ for
$c(\Lambda)\geq 4$ any fixed constant sufficiently large so that Remark 8.6
applies.

Suppose first that $\rho_{i}=0$ for all $i$. By definition of (strong-)cone-
decomposition and Theorem 5.1, necessarily $\theta_{\mathbf{C}}(0)>1$, and so
$\mathrm{sing}M_{i}=\\{0\\}$. We can use Lemma 8.5, Remark 8.6 to obtain bi-
Lipschitz maps $\psi_{i}:B_{1}\to B_{1}$, with $\psi_{i}=0$, which restrict to
$C^{2}$ diffeomorphisms $A_{1,0}\to A_{1,0}$, and satisfy

$\displaystyle\psi_{i}(\overline{M_{1}}\cap B_{1})=\overline{M_{i}}\cap
B_{1},\quad\mathrm{Lip}(\psi_{i})\leq c(\Lambda,M_{1}),$
$\displaystyle\psi_{i}|_{A_{1,1/8}}\text{ is a $(\mathbf{C},\epsilon)$-map.}$

This is a contradiction, and proves Case 2 when $N=1$.

Let us assume now $\rho_{i}>0$ for all $i$. Passing to a further subsequence,
by our inductive hypothesis we can find a $\hat{g}\in\mathcal{G}$,
$\hat{M}\in\mathcal{M}_{7}(B_{1},\hat{g})$,
$\hat{\mathbf{C}}\in\mathcal{C}_{\Lambda}$, and local bi-Lipschitz maps
$\hat{\phi}_{i}:B_{1}\to B_{1}$ such that Theorem 8.1(1)-(5) hold with
$\hat{\phi}_{i}$, $\hat{M}$, $\hat{\mathbf{C}}$, $\eta_{0,\rho_{i}}(M_{i})$,
$C_{\mathcal{S},N-1,\mathcal{G}}$ in place of $\phi_{i}$, $M_{v}$,
$\mathbf{C}_{v}$, $M$, $C_{\mathcal{S},N,\mathcal{G}}$ (respectively).

Let
$\phi_{i}=\eta_{0,\rho_{i}}^{-1}\circ\hat{\phi}_{i}\circ\eta_{0,\rho_{1}}$, so
that $\phi_{i}$ are local bi-Lipschitz maps $B_{\rho_{1}}\to B_{\rho_{i}}$
satisfying

$\displaystyle\phi_{i}(\overline{M_{1}}\cap B_{\rho_{1}})=\overline{M_{i}}\cap
B_{\rho_{i}},\quad\phi_{i}(\mathrm{sing}M_{1}\cap
B_{\rho_{1}})=\mathrm{sing}M_{i}\cap B_{\rho_{i}}$
$\displaystyle\phi_{i}|_{B_{\rho_{1}}\setminus\mathrm{sing}M_{1}}\text{ is a
$C^{2}$ diffeomorphism,}$
$\displaystyle\mathrm{Lip}(\phi_{i}|_{B_{r\rho_{1}}})\leq\rho_{1}^{-1}C_{\mathcal{S},N-1,\mathcal{G}}(r)\text{
for every $r<1$, independent of $i$},$
$\displaystyle\frac{\rho_{1}}{\rho_{i}}\phi_{i}|_{A_{\rho_{1},(1-\sigma)\rho_{1}}}\text{
is a $(\hat{\mathbf{C}},\epsilon)$-map.}$

Define $M_{m\mathbf{C}}$ by

$M_{m\mathbf{C}}=\cup_{j=1}^{m}G_{\mathbf{C}}(\frac{\epsilon_{0}j}{2m}|x|\nu_{\mathbf{C}})\cap
A_{\infty,0},$

so that $M_{m\mathbf{C}}$ looks very close to a multiplicity-$m$ $\mathbf{C}$.
From Lemma 8.5, we can find $C^{2}$ diffeomorphisms
$\hat{\psi}_{i}:A_{1,\rho_{i}/8}\to A_{1,\rho_{i}/8}$ such that

$\displaystyle\hat{\psi}_{i}(M_{m\mathbf{C}}\cap A_{1,\rho_{i}/8})=M_{i}\cap
A_{1,\rho_{i}/8},\quad|D\hat{\psi}_{i}|\leq c(\mathbf{C},m)$
$\displaystyle\hat{\psi}_{i}|_{A_{1,\rho_{i}}}\text{ is a
$(\mathbf{C},\epsilon)$-map.}$

Using the $\hat{f}$ from Lemma 12.1, we can find smooth increasing functions
$f_{i}:[0,1]\to[0,1]$ satisfying

$\displaystyle f_{i}(t)=t\text{ if }t\in[1-2\sigma,1],\quad
f_{i}(t)=\frac{\rho_{i}}{\rho_{1}}t\text{ if }t\in[0,(1+\sigma)\rho_{1}],\quad
0<f_{i}^{\prime}\leq c(\rho_{1}).$

Let $F_{i}(x)=f_{i}(|x|)x/|x|$, and then each $F_{i}$ is a smooth
diffeomorphism $B_{1}\to B_{1}$ which fixes $M_{m\mathbf{C}}\cap B_{1}$,
coincides with the identity in $A_{1,1-\sigma}$, coincides with
$\frac{\rho_{i}}{\rho_{1}}id$ in $B_{(1+\sigma)\rho_{1}}$, and admits the
uniform bound $|DF_{i}|\leq c(\rho_{1})$ independent of $i$.

Define $\psi_{i}=\hat{\psi}_{i}\circ F_{i}\circ\hat{\psi}_{1}^{-1}$. The
$\psi_{i}$ are $C^{2}$ diffeomorphisms $A_{1,\rho_{1}/8}\to A_{1,\rho_{i}/8}$
satisfying:

$\displaystyle\psi_{i}(M_{1}\cap A_{1,\rho_{1}/8})=M_{i}\cap
A_{1,\rho_{i}/8},\quad|D\psi_{i}|_{C^{0}(A_{r,\rho_{1}/4})}\leq
c(M_{1},r)\text{ for }r<1,$ $\displaystyle\psi_{i}|_{A_{1,1-\sigma}}\text{ is
a $(\mathbf{C},\epsilon)$-map},\quad
F_{i}^{-1}\psi_{i}|_{A_{\rho_{1},\rho_{1}/4}}\text{ is a
$(\mathbf{C},\epsilon)$-map.}$

Let $\sigma^{\prime}=10^{-2}\sigma$ and
$r^{\prime}=\frac{1}{1+2\sigma^{\prime}}\rho_{1}$, so that
$A:=A_{(1+\sigma^{\prime})r^{\prime},(1-4\sigma^{\prime})r^{\prime}}\subset
A_{\rho_{1},(1-\sigma)\rho_{1}}$. By (106), (107), and Theorem 8.1(5), we have

$d_{H}(\hat{\mathbf{C}}\cap A,\mathbf{C}\cap A)\leq
2(\epsilon+2\beta)r^{\prime},$

and hence by Theorem 5.1 we can take
$\epsilon(\Lambda,\sigma),\beta(\Lambda,\sigma)$ small to deduce

(108)
$\displaystyle|X_{\mathbf{C}}-X_{\hat{\mathbf{C}}}|_{C^{1}(B_{2\epsilon_{1}}(\mathbf{C})\cap
A_{1,1/2})}\leq\delta_{1}(\Lambda,\sigma^{\prime}),$

with $\epsilon_{1}$, $\delta_{1}$ being the constants from Lemma 8.9.

Write $\tilde{M}_{i}=\frac{\rho_{1}}{\rho_{i}}M_{i}$. On $A$, we have that
$F_{i}^{-1}\psi_{i}=\frac{\rho_{1}}{\rho_{i}}\circ\hat{\psi}_{i}\circ\frac{\rho_{i}}{\rho_{1}}\circ\hat{\psi}_{i}^{-1}$,
and therefore

(109) $\displaystyle F_{i}^{-1}\psi_{i}(\tilde{M}_{1}\cap A)=\tilde{M}_{i}\cap
A,\quad|D(F_{i}^{-1}\psi_{i})|_{C^{0}(A)}\leq c(M_{1}),$ (110) $\displaystyle
F_{i}^{-1}\psi_{i}|_{A}\text{ preserves the flowlines of $X_{\mathbf{C}}$},$
(111) $\displaystyle F_{i}^{-1}\psi_{i}|_{A}\equiv id\text{ outside
$B_{\epsilon r^{\prime}}(\mathbf{C})\subset
B_{\epsilon_{1}r^{\prime}/10}(\mathbf{C})$},$

ensuring $\epsilon(\Lambda)$ is small. Likewise, on $A$ we have
$F_{i}^{-1}\phi_{i}=\rho_{1}\circ\hat{\phi}_{i}\circ\frac{1}{\rho_{1}}$, and
so (109), (110), (111) hold for $F_{i}^{-1}\phi_{i}$ in place of
$F_{i}^{-1}\psi_{i}$ and
$C_{\mathcal{S},N-1,\mathcal{G}}(\frac{1+\sigma}{1+2\sigma})$ in place of
$c(M_{1})$.

In light of (106), (107) (being scale-invariant), (108), and the previous
paragraph, we can therefore apply Lemma 8.9 to the maps $F_{i}^{-1}\psi_{i}$,
$F_{i}^{-1}\phi_{i}$ at scale $B_{r^{\prime}}$ (with $\sigma^{\prime}$ in
place of $\sigma$) to obtain a $C^{2}$ diffeomorphism $h$ on
$A_{\rho_{1},(1-\sigma)\rho_{1}}$ taking $\tilde{M}_{1}=M_{1}$ to
$\tilde{M}_{i}$, and satisfying

$h|_{A_{\rho_{1},r^{\prime}}}=F_{i}^{-1}\psi_{i},\quad
h|_{A_{(1-3\sigma^{\prime})r^{\prime},(1-\sigma)\rho_{1}}}=F_{i}^{-1}\phi_{i},\quad|Dh|\leq
c(\Lambda,\sigma,M_{1},C_{1}).$

If we now define the local bi-Lipschitz maps $\Psi_{i}:B_{1}\to B_{1}$ by

$\Psi_{i}=\left\\{\begin{array}[]{l l}\psi_{i}(x)&x\in A_{1,\rho_{1}}\\\
F_{i}h(x)&x\in A_{\rho_{1},(1-\sigma)\rho_{1}}\\\ \phi_{i}(x)&x\in
B_{(1-\sigma)\rho_{1}}\end{array}\right.,$

then it follows that Theorem 8.1(1)-(4) hold with $\Psi_{i}$, $M_{1}$,
$\mathbf{C}$, $M_{i}$,
$c(\Lambda,\sigma,M_{1},C_{\mathcal{S},N-1,\mathcal{G}},r)$ in place of
$\phi$, $M_{v}$, $\mathbf{C}_{v}$, $M$, $C_{\mathcal{S},N,\mathcal{G}}$. This
contradicts our choice of $M_{i}$, and completes the proof of Case 2. ∎

##  9\. Global finiteness

In this section we prove Theorem 2.4. Here we shall work in $(N^{8},g)$, for
$N^{8}$ a closed Riemannian $8$-manifold with $C^{3}$ metric $g$. We need to
extend some of our definitions to be adapted to global geometry. Recall that
if $M$ is an embedded $C^{2}$ hypersurface of $(N,g)$, then $T^{\perp_{g}}M$
denotes the normal bundle of $M$ in $(N,g)$, and $\nabla^{g}$ is the
connection induced by $g$ on $T^{\perp_{g}}M$. We identity each
$T_{x}^{\perp_{g}}M$ with a subspace of $T_{x}N$ in the obvious fashion. We
write $C^{k}(M,M^{\perp})$ for the space of $C^{k}$ sections of this bundle,
and define $|u|_{C^{k}(N,g)}$ in terms of $\nabla^{g}$. Given $u\in
C^{k}(M\cap U,M^{\perp_{g}})$, write
$\mathrm{graph}_{M,g}(u)=\\{\exp_{N}(u(x)):x\in U\cap M\\}$. We define
convergence of hypersurfaces $M_{i}\to M$ in the same way as in
$\mathbb{R}^{8}$, except with the notion of $\mathrm{graph}_{M,g}$ defined
here.

In direct analogy with the set up of Section 4, a $7$-varifold $V$ in $N$ is
now a measure on the Grassmann bundle $G_{7}(TN)$. We define
$\mathcal{ISV}_{7}(N,g)$ to be the set of stationary integral $7$-varifolds
$V$ in $(N,g)$ with $\mathrm{sing}V$ consisting of a discrete collection of
points. The various special cases and constructions (e.g. $[M]_{g}$,
$f_{\sharp}V$, $\delta_{g}V$, etc.) extend in an obvious fashion to general
$(N,g)$. Lemmas 4.1, 4.2, 4.4 continue to hold for $(N,g)$ in place of
$(U,g)$, but if one prefers one could break $N$ into finitely-many coordinate
neighborhoods, and apply the Lemmas in each one. Let us write
$\theta_{V,g}(x,r)=\omega_{7}^{-1}r^{-7}\mu_{V}(B_{r}^{g}(x))$ for the density
ratio of $V$ w.r.t. $g$-geodesic balls, and if $M$ is a $C^{1}$ embedded
hypersurface, $\theta_{M,g}(x,r)=\theta_{[M]_{g},g}(x,r)$.

###### Proof of Theorem 2.4.

One could prove this Theorem by first constructing a closed analogue of a cone
decomposition, as we did with Theorem 2.1, but since we are only interested in
the parameterization it will be less cumbersome to prove it directly by
contradiction.

We first make a general note about minimal surfaces in $(N,g)$. Since $N$ is
closed, we can find numbers $R(N,g),\eta(N,g)\in(0,1]$ so that if $g^{\prime}$
is any $C^{2}$ metric on $N$ satisfying $|g-g^{\prime}|_{C^{2}(N,g)}\leq\eta$,
then every ball $(B_{R}(x),g^{\prime})$ is isometric to some
$(B_{R}(0)\subset\mathbb{R}^{8},g^{\prime}_{x})$, for $g^{\prime}_{x}$ a
$C^{2}$ metric on $B_{R}(0)$ satisfying
$|g^{\prime}_{x}-g_{\mathrm{eucl}}|_{C^{2}(B_{R}(0))}\leq\min\\{c_{0}^{-1},10^{-2}\\}$,
$c_{0}$ as in (18). In particular, for $V\in\mathcal{ISV}_{7}(N,g^{\prime})$
we have the bounds

$\theta_{V,g^{\prime}}(x,r)\leq 2\theta_{V,g^{\prime}}(x,R)\leq
c_{2}(N,g)\mu_{V}(N)\quad\forall x\in N,r\in(0,R).$

So by (19),

(112) $V\neq
0\iff\mu_{V}(N)\geq\frac{1}{2c_{2}}\iff\mathrm{spt}V\neq\emptyset,$

and

(113)
$\mu_{V}(N)\leq\Lambda\implies\theta_{V,g^{\prime}}(x,r)\leq\Lambda^{\prime}:=c_{2}\Lambda\quad\forall
x\in N,\forall r<R.$

Let $\sigma=\frac{1}{100(1+I)}$, and ensure
$\epsilon\leq\min\\{\epsilon_{1}(\Lambda^{\prime}),10^{-2}\epsilon_{0}(\Lambda^{\prime})\\}$,
for $\epsilon_{1}$ as in Lemma 8.9 and $\epsilon_{0}$ as in the discussion of
Section 8. We will choose constants $\delta^{\prime},\beta,\epsilon>0$ as we
progress, along with several choices, which a posteriori can all be fixed, but
to guide the reader our dependencies look like the following ($\to$ meaning
“depends on”):

$i_{0}\to\epsilon_{M}\to\\{B_{r_{\alpha}}(y_{\alpha})\\}_{\alpha}\to
M,\delta^{\prime},\beta,\epsilon\quad\text{and}\quad\delta^{\prime}\to\beta\to\epsilon\to\sigma,\Lambda^{\prime},I.$

Suppose the theorem fails. Then for every $i$ large there is a set of $C^{3}$
metric $\mathcal{G}_{i}$ on $N$ satisfying $|g^{\prime}-g|_{C^{3}(N,g)}\leq
1/i$ for all $g^{\prime}\in\mathcal{G}_{i}$, such that we can find sequences
$\\{g_{ij}\\}_{j}\subset\mathcal{G}_{i}$,
$\\{M_{ij}\in\mathcal{M}_{7}(N,g_{ij})\\}_{j}$, satisfying
$\mathcal{H}^{7}_{g_{ij}}(M_{ij})\leq\Lambda$,
$\mathrm{index}(M_{ij},N,g_{ij})\leq I$, with the property that for every
$j^{\prime}>j$ there does not exist a bi-Lipschitz mapping
$\phi:(N,g)\to(N,g)$ satisfying Theorem 2.4(1)-(2)-(3) with $M_{ij}$,
$M_{ij^{\prime}}$, $j^{\prime}$ in place of $M_{v}$, $M$, $C$. As before, this
contradiction hypothesis is preserved if we replace $j$ with any subsequence
of $j$.

If $M_{ij}=M_{ij^{\prime}}=\emptyset$ then trivially such a $\phi$ exists, so
by (112) after discarding finitely-many terms in each sequence we must have
$\mathcal{H}^{7}_{g_{i}}(M_{ij})\geq\frac{1}{2c_{2}}$ for all $i$, $j$. After
passing to subsequences in $j$, we can assume there are $C^{2}$ metrics
$g_{i}$ (satisfying $|g_{i}-g|_{C^{2}(N,g)}\leq 1/i$) so that $g_{ij}\to
g_{i}$ in $C^{2}(N,g)$, and (by Lemma 4.4) there are non-zero varifolds
$V_{i}\in\mathcal{ISV}_{7}(N,g_{i})$, and discrete sets $\mathcal{I}_{i}$
consisting of at most $I$ points, so that $[M_{ij}]_{g_{ij}}\to V_{i}$ as
varifolds in $(N,g)$, and in $C^{2}$ on compact subsets of
$N\setminus(\mathrm{sing}V_{i}\cup\mathcal{I}_{i})$. Moreover, we have
$\mu_{V_{i}}(N)\leq\Lambda$, and $\mathrm{index}(V_{i},N,g_{i})\leq I$.

Trivially $g_{i}\to g$ in $C^{2}(N,g)$, so passing to a subsequence in $i$ we
find a non-zero $V\in\mathcal{ISV}_{7}(N,g)$, and at most $I$ points
$\mathcal{I}$, so that $V_{i}\to V$ as varifolds in $(N,g)$, and in $C^{2}$ on
compact subsets of $N\setminus(\mathrm{sing}V\cup\mathcal{I})$. Moreover,
since we cannot have points $x_{i}\in\mathrm{sing}V_{i}$ converging to a point
in $\mathrm{reg}V\setminus\mathcal{I}$, and $N$ is compact, after enlarging
$\mathcal{I}$ to consist of possibly $2I$ points, we can assume that for every
$\tau>0$ there is an $i_{1}$ so that
$\mathrm{sing}V_{i}\cup\mathcal{I}_{i}\subset
B_{\tau}^{g}(\mathrm{sing}V\cup\mathcal{I})$ for all $i>i_{1}$. $V$ satisfies
$\mu_{V}(N)\leq\Lambda$, $\mathrm{index}(V,N,g)\leq I$.

Write
$\mathrm{sing}V\cup\mathcal{I}=\\{y_{1},\ldots,y_{d}\\}\subset\mathrm{spt}V$,
and take $\delta^{\prime},\epsilon,\beta>0$ (for the moment) arbitrary. By
working in normal coordinates centered at each $y_{\alpha}$, by Lemmas 4.1,
4.4, (18), (113), Theorem 6.3, Remark 6.4, and the maximum principle [Ilm96],
we can choose radii $r_{\alpha}\in(0,1/8)$, integers $m_{\alpha}$, and cones
$\mathbf{C}_{\alpha}\in\mathcal{C}$, satisfying
$m_{\alpha}\theta_{\mathbf{C}_{\alpha}}(0)\leq\Lambda^{\prime}$, so that:

  1. A.1

the balls $\\{B^{g}_{8r_{\alpha}}(y_{\alpha})\\}_{\alpha}$ are disjoint, and
each $(B^{g}_{8r_{\alpha}}(y_{\alpha}),g)$ is isometric via normal coordinates
to (and will henceforth be identified with)
$(B_{8r_{\alpha}}(0)\subset\mathbb{R}^{8},g_{\alpha})$ for some $C^{3}$ metric
$g_{\alpha}$ satisfying
$|g_{\alpha}-g_{\mathrm{eucl}}|_{C^{3}(B_{8r_{\alpha}}(0),g_{\mathrm{eucl}})}\leq\delta^{\prime}/2$;

  2. A.2

$\mathrm{spt}V\cap(A_{8r_{\alpha},0}(0),g_{\alpha})$ is a
$(\mathbf{C}_{\alpha},1,\beta/2)$-strong-cone region;

  3. A.3

for each $\alpha$, we have

(114) $\displaystyle d_{H}(\mathrm{spt}V\cap
B_{r_{\alpha}},\mathbf{C}_{\alpha}\cap
B_{r_{\alpha}})\leq(\delta^{\prime}/2)r_{\alpha}$ (115)
$\displaystyle\theta_{V}(0,r_{\alpha}/2)\geq(m_{\alpha}-1/4)\theta_{\mathbf{C}_{\alpha}}(0),\quad\theta_{V}(0,r_{\alpha})\leq(m_{\alpha}+1/4)\theta_{\mathbf{C}_{\alpha}}.$

Let $O=N\setminus\cup_{\alpha}B_{2r_{\alpha}}(y_{\alpha})$,
$O^{\prime}=N\setminus\cup_{\alpha}B_{r_{\alpha}}(y_{\alpha})$, and then we
can write $V\llcorner O^{\prime}=n_{1}[S_{1}]+\ldots+n_{p}[S_{p}]$ for
$n_{s}\in\mathbb{N}$, and the $S_{s}$ being a $C^{2}$, disjoint, closed (as
sets), embedded minimal hypersurface in $(O^{\prime},g)$. For each $S_{s}$,
define the map $G_{s}:T^{\perp_{g}}S_{s}\to N$ by
$G_{s}(x,v)=\exp_{N}|_{x}(v)$. By A.2, we can choose an
$0<\epsilon_{M}\leq\epsilon\min\\{1,\min_{\alpha}r_{\alpha}\\}$ so that each
$G_{s}$ is a $C^{2}$ diffeomorphism from $\\{(x,v)\in
T^{\perp_{g}}S_{s}:|v|<4\epsilon_{M}\\}$ onto its image $O_{s}$, and
$O_{s}\supset B^{g}_{4\epsilon_{M}}(\mathrm{spt}V)\cap O$, and $G_{s}$ admits
the bound $|DG_{s}|\leq 2$, and all the images $O_{s}$ are disjoint.

Let us ensure $\delta^{\prime}(I,\Lambda^{\prime},\beta,\sigma)$ is smaller
than the constant $\delta_{l,I}(l,I,\gamma=\beta,\sigma)$ from Theorem 7.1,
for $l$ chosen so that $\tilde{\theta}_{l}\geq\Lambda^{\prime}$. We can then
fix an $i_{0}$, depending on
$\delta^{\prime},\epsilon_{M},\beta,V,\\{B_{r_{\alpha}}(y_{\alpha})\\}_{\alpha}$,
and pass to a subsequence in $j$, so that for each $j$ we have:

  1. B.1

$M_{i_{0}j}\subset B^{g}_{\epsilon_{M}}(\mathrm{spt}V)$;

  2. B.2

in each $(B_{8r_{\alpha}}(0),g_{\alpha})$, the metric $g_{i_{0}j}$ satisfies
$|g_{i_{0}j}-g_{\mathrm{eucl}}|_{C^{3}(B_{8r_{\alpha}}(0),g_{\mathrm{eucl}})}\leq\delta^{\prime}$;

  3. B.3

for every $\alpha$, $M_{i_{0}j}\cap(A_{8r_{\alpha},r_{\alpha}/8},g_{\alpha})$
is a $(\mathbf{C}_{\alpha},m_{\alpha},\beta)$-strong-cone region;

  4. B.4

we have for every $\alpha$:

(116) $\displaystyle d_{H}(M_{i_{0}j}\cap
B_{r_{\alpha}},\mathbf{C}_{\alpha}\cap
B_{r_{\alpha}})\leq\delta^{\prime}r_{\alpha}$ (117)
$\displaystyle\theta_{M_{i_{0}j},g_{i_{0}j}}(0,r_{\alpha}/2)\geq(m_{\alpha}-1/2)\theta_{\mathbf{C}_{\alpha}}(0),\quad\theta_{M_{i_{0}j},g_{i_{0}j}}(0,r_{\alpha})\leq(m_{\alpha}+1/2)\theta_{\mathbf{C}_{\alpha}};$

  5. B.5

for each $s=1,\ldots,p$, there are $C^{2}$ functions $\\{u_{jsk}:S_{s}\to
S_{s}^{\perp_{g}}\\}_{k=1}^{n_{s}}$ so that

$M_{i_{0}j}\cap O_{s}=\cup_{k=1}^{n_{s}}\\{G_{s}(x,u_{jsk}(x)):x\in
S_{s}\\},\quad|u_{jsk}|_{C^{2}(S_{s})}\leq\epsilon_{M}.$

Having fixed this $i_{0}$, let us for simplicity drop the $i$ notation, think
only of the sequence $M_{j}\equiv M_{i_{0}j}$ and $g_{j}\equiv g_{i_{0}j}$. We
can forget all the other sequences. We shall build bi-Lipschitz maps
$\phi_{j}:(N,g)\to(N,g)$ with uniform Lipschitz constant taking
$\overline{M_{1}}\to\overline{M_{j}}$, which will contradict our choice of
$M_{j}$.

By B.2, B.4, and since every $r_{\alpha}\leq 1$, we can apply Theorem 7.1 to
each $[M_{j}]_{g_{j}}$ in each ball $(B_{r_{\alpha}},g_{j})$, to deduce there
is a finite collection of $(\Lambda^{\prime},\sigma,\beta)$-smooth regions
$\mathcal{S}$ and an integer $K$ so that for every $\alpha$ and $j$, there is
a radius $r_{\alpha j}\in(\frac{4}{5}r_{\alpha},r_{\alpha})$ so that
$M_{j}\cap(B_{r_{\alpha j}},g_{j})$ admits a
$(\Lambda^{\prime},\beta,\mathcal{S},K)$-strong-cone decomposition. From B.3,
we deduce that each $M_{j}\cap(B_{2r_{\alpha}},g_{j})$ admits a
$(\Lambda^{\prime},\beta,\mathcal{S},K+1)$-strong-cone decomposition.

Assuming $\beta(\Lambda^{\prime},\epsilon,\sigma)$ is small, we can apply
Theorem 8.1, pass to a subsequence in $j$, and obtain bi-Lipschitz maps
$\phi_{\alpha j}:B_{2r_{\alpha}}\to B_{2r_{\alpha}}$, cones
$\\{\hat{\mathbf{C}}_{\alpha}\\}_{\alpha}\subset\mathcal{C}_{\Lambda}^{\prime}$,
and an increasing function $C:[0,1)\to\mathbb{R}$ (independent of $\alpha,j$)
satisfying the properties:

  1. C.1

$\phi_{\alpha j}(\overline{M_{1}}\cap B_{2r_{\alpha}})=\overline{M_{j}}\cap
B_{2r_{\alpha}}$, and $\phi_{\alpha j}(\mathrm{sing}M_{1}\cap
B_{2r_{\alpha}})=\mathrm{sing}M_{j}\cap B_{2r_{\alpha}}$;

  2. C.2

$\phi_{\alpha j}|_{B_{2r_{\alpha}}\setminus\mathrm{sing}M_{1}}$ is a $C^{2}$
diffeomorphism;

  3. C.3

$\mathrm{Lip}_{g_{\alpha}}(\phi_{\alpha j}|_{B_{2r_{\alpha}\rho}})\leq
C(\rho)$ for every $\rho<1$;

  4. C.4

$\phi_{\alpha j}|_{A_{2r_{\alpha},2r_{\alpha}(1-\sigma)}}$ is a
$(\hat{\mathbf{C}}_{\alpha},\epsilon)$-map;

  5. C.5

$d_{H}(\hat{\mathbf{C}}_{\alpha}\cap
A_{2r_{\alpha},2r_{\alpha}(1-\sigma)},M_{j}\cap
A_{2r_{\alpha},2r_{\alpha}(1-\sigma)})\leq 2r_{\alpha}\epsilon$ and
$\mathrm{sing}M_{j}\cap B_{2r_{\alpha}}\subset B_{2r_{\alpha}(1-\sigma)}$.

We now work towards gluing these $\phi_{\alpha j}$ together.

We first build our large-scale diffeomorphisms. Define
$g_{sj}:T^{\perp_{g}}S_{s}\to T^{\perp_{g}}S_{s}$ as follows. Given any $x\in
S_{s}$, there is a neighborhood $U\subset S_{s}$ containing $x$, in which the
normal bundle $T^{\perp_{g}}(U\cap S_{s})$ is isometric to a trivialization
$(U\cap S_{s})\times\mathbb{R}\ni(x,t)$. Under this identification, we can
think of the $u_{jsk}$ as taking values in $(-4\epsilon_{M},4\epsilon_{M})$,
and so by the maximum principle and the smallness of the singular set of
$M_{j}$ we can assume

$u_{js1}<\ldots<u_{jsm_{s}}\quad\forall j=1,2,\ldots,\forall s=1,\ldots,p.$

Let $f_{\\{a_{j}\\}_{j},\\{b_{j}\\}_{j}}(t)$ be the function from Lemma 12.1
with $m_{s}$ in place of $k$ and $\epsilon_{M}$ in place of $\epsilon$. For
$(x,t)\in(U\cap S_{j})\times\mathbb{R}\cong T^{\perp_{g}}(U\cap S_{j})$, we
define

$g_{sj}(x,t)=(x,f_{\\{u_{1sk}(x)\\}_{k},\\{u_{jsk}(x)\\}_{k}}(t)),$

so that $g_{sj}(x,u_{1sk}(x))=(x,u_{jsk}(x))$ for every $k=1,2,\ldots,m_{j}$.
Trivially, $g_{sj}$ is a $C^{2}$ diffeomorphism $(U\cap
S_{s})\times(-4\epsilon_{M},4\epsilon_{M})$ which coincides with the identity
for $|t|\geq 2\epsilon_{M}$. By Remark 12.1, this definition is independent of
our choice of trivialization, and so gives rise a diffeomorphism from
$T^{\perp_{g}}S_{s}$ to itself. By Item A.3, the number

$D=\inf\\{|u_{1sk}(x)-u_{1sk^{\prime}}(x)|:x\in S_{s},k\neq
k^{\prime},s=1,\ldots,p\\}>0,$

and hence by Lemma 12.1 we have the uniform $C^{1}$ bound $|Dg_{sj}|\leq
c(N,V,M_{1})$.

Now define $\psi_{j}:(O,g)\to(N,g)$ by setting

$\psi_{j}(x)=\left\\{\begin{array}[]{l l}G_{s}^{-1}\circ g_{sj}\circ
G_{s}&x\in O_{s}\\\ x&x\not\in O_{s}\end{array}\right..$

By our construction and by B.1, $\psi_{j}$ is a well-defined $C^{2}$
diffeomorphism onto its image, satisfying

$\psi_{j}(M_{1}\cap O)=M_{j}\cap\psi_{j}(O),\quad|\psi_{j}(x)-x|\leq
10\epsilon_{M},\quad|D\psi_{j}|\leq c(N,V,M_{1}).$

For each $\alpha$, set $S_{\alpha}^{\prime}=\mathrm{spt}V\cap
A_{8r_{\alpha},r_{\alpha}}$. Let $\nu_{\alpha}$ be a choice of unit normal for
$S_{\alpha}^{\prime}$. By A.2, Theorem 5.1 and our choice of $\epsilon_{M}$,
provided $\delta^{\prime}(\Lambda^{\prime}),\beta(\Lambda^{\prime})$ are
sufficiently small, the function
$G_{\alpha}:S_{\alpha}^{\prime}\times(-4\epsilon_{M},4\epsilon_{M})\to\mathbb{R}^{8}$
given by $G_{\alpha}(x,v)=\exp_{g_{\alpha}}|_{x}(t\nu_{\alpha}(x))$ is a well-
defined $C^{2}$ diffeomorphism onto its image. If we let
$X_{\alpha}:=DG_{\alpha}(\partial_{t})$, then after shrinking
$\delta^{\prime}(\Lambda^{\prime}),\beta(\Lambda^{\prime})$ and flipping
$\nu_{\alpha}$ as necessary, $X_{\alpha}$ is a $C^{2}$ vector field on
$B_{3\epsilon_{M}}(\mathbf{C}_{\alpha})\cap A_{4r_{\alpha},2r_{\alpha}}$
satisfying

$|X_{\alpha}-X_{\mathbf{C}_{\alpha}}|+r_{\alpha}|DX_{\alpha}-X_{\mathbf{C}_{\alpha}}|\leq
c(\Lambda^{\prime})(\beta+\delta^{\prime}).$

Ensuring $\beta(\Lambda^{\prime},\epsilon)$,
$\delta^{\prime}(\Lambda^{\prime},\epsilon)$ are small, and recalling our
restriction $\epsilon_{M}\leq\epsilon r_{\alpha}$, we can argue as in Lemma
8.4 and obtain $C^{2}$ diffeomorphisms $\psi_{\alpha
j}:(A_{4r_{\alpha},r_{\alpha}},g_{\alpha})\to(B_{8r_{\alpha}},g_{\alpha})$
such that

(118) $\displaystyle\psi_{\alpha j}(M_{1}\cap
A_{4r_{\alpha},r_{\alpha}})=M_{j}\cap\psi_{\alpha
j}(A_{4r_{\alpha},r_{\alpha}}),\quad|D\psi_{\alpha j}|\leq c(N,V,M_{1}),$
(119) $\displaystyle\psi_{\alpha
j}|_{A_{4r_{\alpha},3r_{\alpha}}}=\psi_{j},\quad\psi_{\alpha
j}|_{A_{2r_{\alpha},r_{\alpha}}}\text{ is a
$(\mathbf{C}_{\alpha},\epsilon)$-map}.$

We use the $\psi_{\alpha j}$ to transition from our large-scale $\psi_{j}$,
and the small-scale maps $\phi_{\alpha j}$ arising from the cone
decompositions of $M_{i}$ in each ball $B_{2r_{\alpha}}$.

Let $\sigma^{\prime}=10^{-2}\sigma$, and
$r_{\alpha}^{\prime}=(1+5\sigma^{\prime})^{-1}r_{\alpha}$. Then we have

$A_{\alpha}:=A_{2r_{\alpha}^{\prime}(1+\sigma^{\prime}),2r_{\alpha}^{\prime}(1-4\sigma^{\prime})}\subset
A_{2r_{\alpha},2r_{\alpha}(1-\sigma)}.$

A.3, C.4, C.5 imply that $d_{H}(\mathbf{C}_{\alpha}\cap
B_{1},\hat{\mathbf{C}}_{\alpha}\cap B_{1})\leq
c(\Lambda^{\prime})(\epsilon+\beta)$, and so by ensuring
$\beta(\Lambda^{\prime},\sigma),\epsilon(\Lambda^{\prime},\sigma)$ are
sufficiently small Theorem 5.1 implies that both $X_{\mathbf{C}_{\alpha}}$ and
$X_{\hat{\mathbf{C}}_{\alpha}}$ are defined on
$B_{2\epsilon_{1}r_{\alpha}^{\prime}}(\mathbf{C}_{\alpha})\cap A_{\alpha}$ and
satisfy

$|X_{\mathbf{C}_{\alpha}}-X_{\hat{\mathbf{C}}_{\alpha}}|+r_{\alpha}^{\prime}|DX_{\mathbf{C}_{\alpha}}-X_{\hat{\mathbf{C}}_{\alpha}}|\leq\delta_{1}(\Lambda^{\prime},\sigma^{\prime}).$

Here $\epsilon_{1}(\Lambda^{\prime})$,
$\delta_{1}(\Lambda^{\prime},\sigma^{\prime})$ are the constants from Lemma
8.9.

Combined with (118), (119), C.3, C.4, B.3, and our choice of $A_{\alpha}$, we
can therefore apply Lemma 8.9 (at scale $B_{2r_{\alpha}^{\prime}}$ and with
$\sigma^{\prime},\epsilon_{1}$ in place of $\sigma,\epsilon$) to obtain
$C^{2}$ diffeomorphisms $\hat{\phi}_{\alpha
j}:(A_{\alpha},g_{\alpha})\to(A_{\alpha},g_{\alpha})$ satisfying

$\displaystyle\hat{\phi}_{\alpha j}(M_{1}\cap A_{\alpha})=M_{j}\cap
A_{\alpha},\quad|D\hat{\phi}_{\alpha j}|\leq c(N,V,M_{1}),$
$\displaystyle\hat{\phi}_{\alpha
j}|_{A_{2r_{\alpha}^{\prime}(1+\sigma^{\prime}),2r_{\alpha}^{\prime}}}=\psi_{\alpha
j},\quad\hat{\phi}_{\alpha
j}|_{A_{2r_{\alpha}^{\prime}(1-3\sigma^{\prime}),2r_{\alpha}^{\prime}(1-4\sigma^{\prime})}}=\phi_{\alpha
j}.$

We now define $\phi_{j}$ by setting

$\phi_{j}(x)=\left\\{\begin{array}[]{l l}\psi_{j}(x)&x\in
N\setminus\cup_{\alpha}B_{3r_{\alpha}}(y_{\alpha})\\\ \psi_{\alpha j}(x)&x\in
A_{4r_{\alpha},2r_{\alpha}^{\prime}}\\\ \hat{\phi}_{\alpha j}(x)&x\in
A_{\alpha}\\\ \phi_{\alpha j}(x)&x\in
B_{2r_{\alpha}^{\prime}(1-3\sigma^{\prime})}\end{array}\right.$

By B.1 and our construction, the $\phi_{j}$ are bi-Lipschitz maps
$(N,g)\to(N,g)$ satisfying

$\displaystyle\phi_{j}(\overline{M_{1}})=\overline{M_{j}},\quad\phi_{j}(\mathrm{sing}M_{1})=\mathrm{sing}M_{j},$
$\displaystyle\phi_{j}|_{N\setminus\mathrm{sing}M_{1}}\text{ is a $C^{2}$
diffeomorphism},\quad\mathrm{Lip}_{g}(\phi_{j})\leq c(N,V,M_{1}).$

This is a contradiction, and finishes the proof of Theorem 2.4. ∎

##  10\. Other corollaries

Here we prove Theorems 2.1 and 2.6. Theorem 2.1 is a trivial consequence of
Theorems 7.1, 8.1, but we include a proof for the sake of precision. We work
in $\mathbb{R}^{8}$.

###### Proof of Theorem 2.1.

We first note that the existence of $V$, $\mathcal{I}$, and convergence
$M_{i}\to V$, follows directly from Lemma 4.4.

A straightforward contradiction argument, using Theorem 5.1, Lemma 4.4, and
the constancy theorem implies that provided $\delta(\Lambda,I,\sigma)$ is
sufficiently small, we can find integers $m_{i}\in\\{0,1,\ldots\\}$ so that
$m_{i}\theta_{\mathbf{C}}(0)\leq\Lambda$, and

$\theta_{[M_{i}]_{g_{i}}}(0,1-\sigma)\leq(m_{i}+1/2)\theta_{\mathbf{C}}(0),\quad\theta_{[M_{i}]_{g_{i}}}(0,(1-\sigma)/2)\geq(m_{i}-1/2)\theta_{\mathbf{C}}(0).$

Passing to a subsequence, there no loss in assuming $m_{i}\equiv m$. If $m=0$,
then provided we take $\delta^{\prime}(\Lambda,\sigma)$ small (19) implies
$M_{i}\cap B_{1-2\sigma}=\emptyset$ for all $i$ (and hence
$\mu_{V}(B_{1-2\sigma})=\emptyset$), so we can just take $r=1-2\sigma$,
$\phi_{i}\equiv id$.

By Theorem 7.1, choosing $\delta^{\prime}(\Lambda,I,\beta,\sigma)$ small, we
can find a finite collection $\mathcal{S}$ of $(\Lambda,\beta,5\sigma)$-smooth
models, and a constant $K(\Lambda,I,\beta,\sigma)$, so that for all $i>>1$
there are radii $r_{i}\in(1-200\sigma(I+1),1)$ so that
$M_{i}\cap(B_{r_{i}}(0),g_{i})$ admits a $(\Lambda,\beta,\mathcal{S},K)$-cone
decomposition. Passing to a subsequence, we can assume $r_{i}\to(1+\sigma)r$.

Trivially $\mathcal{S}$ are also $(\Lambda,\beta,3\sigma)$-smooth models, and
so for every $i>>1$ and $r^{\prime}\in((1-\sigma)r,(1+\sigma)r)$, each
$M_{i}\cap(B_{r^{\prime}}(0),g_{i})$ admits a
$(\Lambda,\beta,\mathcal{S},K)$-cone decomposition (though not explicitly
stated, the outer-most region in the decomposition of Theorem 7.1 is always a
smooth region). This proves the first assertion.

To prove existence of the $\phi_{i}$, we apply Theorem 8.1 at scale
$B_{(1+\sigma)r}(0)$ (with $3\sigma$ in place of $\sigma$, $\epsilon>0$
arbitrary), and deduce that after passing to a further subsequence, we can
find a constant $C$, a cone $\mathbf{C}\in\mathcal{C}$, and local bi-Lipschitz
maps $\phi_{i}:B_{(1+\sigma)r}\to B_{(1+\sigma)r}$ satisfying

(120) $\displaystyle\phi_{i}(\overline{M_{1}}\cap
B_{(1+\sigma)r})=\overline{M_{i}},\quad\phi_{i}(\mathrm{sing}M_{1}\cap
B_{(1+\sigma)r})=\mathrm{sing}M_{i},$ (121)
$\displaystyle\mathrm{Lip}(\phi_{i}|_{B_{r}})\leq C,$ (122)
$\displaystyle\phi_{i}|_{A_{(1+\sigma)r,(1-2\sigma)r}}\text{ is a
$(\mathbf{C},\epsilon)$-map},\quad\mathrm{sing}M_{i}\subset B_{(1-2\sigma)r}.$

(122) implies $|\phi_{i}(x)|=|x|$ for $x\in A_{(1+\sigma)r,(1-2\sigma)r}$, and
so we can restrict $\phi_{i}|_{B_{r}}$ to obtain our required maps.

Passing to one last subsequence, we can assume there is a Lipschitz
$\phi_{\infty}:B_{r}\to B_{r}$ so that $\phi_{i}\to\phi_{\infty}$ in
$C^{\alpha}(B_{r})$ for all $\alpha\in(0,1)$. Since $|\phi_{i}(x)|=|x|$ for
$|x|\in((1-2\sigma)r,r)$, we have for all $i$:

(123)
$\max\\{|\phi_{i}^{-1}(x)|,|\phi_{i}(x)|\\}\leq\max\\{(1-2\sigma)r,|x|\\},\quad|\phi_{\infty}(x)|\leq\max\\{(1-2\sigma)r,|x|\\}.$

Varifold convergence $[M_{i}]_{g_{i}}\to V$ in $B_{1}$, (19), and (123) imply
$\phi_{\infty}(\overline{M_{1}}\cap B_{r})\subset\mathrm{spt}V\cap B_{r}$. On
the other hand, again from (19) and varifold convergence, if
$x\in\mathrm{spt}V\cap B_{r}$, then there is a sequence
$x_{i}\in\overline{M_{1}}\cap B_{r}$ so that $\phi_{i}(x_{i})\to x$. By (123)
and since $\overline{M_{1}}$ is closed, there is no loss in assuming that
$x_{i}\to x^{\prime}\in\overline{M_{1}}\cap B_{r}$. The uniform Lipschitz
bound (121) then implies $\phi_{\infty}(x^{\prime})=x$. This proves
$\phi_{\infty}(\overline{M_{1}}\cap B_{r})=\mathrm{spt}V\cap B_{r}$. ∎

###### Proof of Theorem 2.6.

I first claim that given $\beta>0$ there is an $\eta(\Lambda,\beta)$ so that
if $V\in\mathcal{ISV}_{7}(B_{10},g_{\mathrm{eucl}})$ satisfies

(124)
$\theta_{V}(0,10)\leq\Lambda,\quad\theta_{V}(0,10)-\theta_{V}(0,\eta)\leq\eta,\quad\mathrm{index}(V,A_{10,\eta},g_{\mathrm{eucl}})=0,$

then we can find $\mathbf{C}\in\mathcal{C}_{\Lambda}$, $m\in\mathbb{N}$ with
$m\theta_{\mathbf{C}}(0)\leq\Lambda$, and $C^{2}$ functions
$\\{u_{1},\ldots,u_{m}:\mathbf{C}\cap A_{8,1}\to\mathbf{C}^{\perp}\\}$, so
that

(125) $V\llcorner A_{8,1}=\sum_{j=1}^{m}[G_{\mathbf{C}}(u_{j})\cap
A_{8,1}],\quad|u_{j}|_{C^{2}}\leq\beta.$

Suppose otherwise: then there are sequences $\eta_{i}\to 0$, and
$V_{i}\in\mathcal{ISV}_{7}(B_{10},g_{\mathrm{eucl}})$, satisfying (124), but
failing (125) for any $\mathbf{C}\in\mathcal{C}_{\Lambda}$, $m\in\mathbb{N}$
with $m\theta_{\mathbf{C}}(0)\leq\Lambda$. By Lemma 4.4, we can find a
$V\in\mathcal{ISV}_{7}(B_{10},g_{\mathrm{eucl}})$ with
$\theta_{V}(0,10)\leq\Lambda$, $\mathrm{index}(V,B_{10},g_{\mathrm{eucl}})=0$,
so that $M_{i}\to M$ as varifolds in $B_{10}$, and $V_{i}\to V$ smoothly on
compact subsets of $A_{10,0}$. By (17), $V$ is dilation-invariant and stable,
and hence $V=m[\mathbf{C}]\llcorner B_{10}$ for some
$\mathbf{C}\in\mathcal{C}$, and
$m\theta_{\mathbf{C}}(0)=\theta_{V}(0,10)\leq\Lambda$. Smooth convergence then
implies (125) holds for $i>>1$, which is a contradiction. This proves my first
claim.

I next claim that given any $\beta,\delta>0$, and
$M\in\mathcal{M}_{7}(\mathbb{R}^{8},g_{\mathrm{eucl}})$ satisfying (3), then
we can find a positive $R_{M}\in\mathbb{R}$, a
$\mathbf{C}\in\mathcal{C}_{\Lambda}$, and an $m\in\mathbb{N}$, satisfying
$m\theta_{\mathbf{C}}(0)\leq\Lambda$, so that the following is true:

(126) $\displaystyle d_{H}(\mathrm{spt}M\cap B_{R_{M}/2},\mathbf{C}\cap
B_{R_{M}/2})\leq\delta R_{M}/2$ (127)
$\displaystyle(m-1/2)\theta_{\mathbf{C}}(0)\leq\theta_{M}(0,R_{M}/4)\quad\theta_{M}(0,R_{M}/2)\leq(m+1/2)\theta_{\mathbf{C}}(0)$
(128) $\displaystyle M\cap(A_{\infty,R_{M}},g_{\mathrm{eucl}})\text{ is a
$(\mathbf{C},m,\beta)$-strong-cone region.}$

By Lemma 4.4 and (17), we can take $m[\mathbf{C}]$ to be any (the) tangent
cone at infinity for $M$. More precisely, there are radii $R_{i}\to\infty$ so
that $[\eta_{0,R_{i},\sharp}(M)]\to m[\mathbf{C}]$ as varifolds and in the
local Hausdorff distance, and smoothly with multiplicity-$m$ on compact
subsets of $\mathbb{R}^{8}\setminus\\{0\\}$. In particular, (126), (127) hold
for $R=R_{i}$ for any choice of $R_{i}$ large.

Choose $\beta^{\prime}(\Lambda)\leq\beta(\Lambda)$ sufficiently small so that
if $\mathbf{C}^{\prime},\mathbf{C}^{\prime\prime}\in\mathcal{C}_{\Lambda}$
satisfy $d_{H}(\mathbf{C}^{\prime}\cap B_{1},\mathbf{C}^{\prime\prime}\cap
B_{1})\leq 2\beta^{\prime}$, then
$\mathbf{C}^{\prime}\in\mathcal{C}(\mathbf{C}^{\prime\prime})$ (as guaranteed
by Theorem 5.1). Choose $\eta(\Lambda,\beta^{\prime})$ as in Claim 1, and now
taking $R$ sufficiently large so that
$\theta_{M}(0,\infty)-\theta_{M}(0,R)\leq\eta$,
$\mathrm{index}(M,A_{\infty,R},g_{\mathrm{eucl}})=0$, it follows from Claim 1
that $M\cap(A_{\infty,8R/\eta},g_{\mathrm{eucl}})$ is a
$(\mathbf{C},m,\beta^{\prime})$-cone region. It then follows by Theorem 6.3,
provided $\beta^{\prime}(\Lambda,\beta)$ is sufficiently small, that
$M\cap(A_{\infty,R_{i}},g_{\mathrm{eucl}})$ is a
$(\mathbf{C},m,\beta)$-strong-cone region for all $i>>1$. This proves my
second claim.

Let $\sigma=\frac{1}{100(I+1)}$, and let
$2\delta=\delta_{l,I}(l,I,\gamma=\beta,\sigma)$ be the constant from Theorem
7.1, with $l$ chosen so that $\tilde{\theta}_{l}\geq\Lambda$. Let $R_{M}$ be
the radius from Claim 2 with this choice of $\delta$, and $\beta>0$ to be
determined later. It will suffice to prove Theorem 2.6 with
$\lambda=R_{M}^{-1}$.

Suppose, towards a contradiction, Theorem 2.6 fails. Then there is a sequence
of $M_{i}\in\mathcal{M}_{7}(\mathbb{R}^{8},g_{\mathrm{eucl}})$ satisfying (3)
and normalized so that $R_{M_{i}}=1$, and a radius $\rho$, with the property
that for any $i^{\prime}>i$, there does not exist a local bi-Lipschitz
$\phi:\mathbb{R}^{8}\to\mathbb{R}^{8}$ satisfying

$\displaystyle\phi(\overline{M_{i}})=\overline{M_{i^{\prime}}},\quad\phi(\mathrm{sing}M_{i})=\mathrm{sing}M_{i^{\prime}}$
$\displaystyle\phi|_{\mathbb{R}^{8}\setminus\mathrm{sing}M_{i}}\text{ is a
$C^{2}$ diffeomorphism},\quad\mathrm{Lip}(\phi|_{B_{\rho}})\leq i^{\prime}.$

By our normalization, we have for each $i$ an $m_{i}\in\mathbb{N}$,
$\mathbf{C}_{i}\in\mathcal{C}_{\Lambda}$ satisfying
$m_{i}\theta_{\mathbf{C}_{i}}(0)\leq\Lambda$, so that (126), (127), (128) hold
with $M_{i}$, $1$, $m_{i}$, $\mathbf{C}_{i}$ in place of $M$, $R_{M}$, $m$,
$\mathbf{C}$. By Theorem 5.1, after passing to a subsequence, we can assume
there is a fixed $\mathbf{C}\in\mathcal{C}_{\Lambda}$, $m\in\mathbb{N}$ (with
$m\theta_{\mathbf{C}}(0)\leq\Lambda$), so that (126), (127), (128) hold with
$M_{i}$, $1$, $2\delta$, $2\beta$ in place of $M$, $R_{m}$, $\delta$, $\beta$.

By our choice of $\delta$, we can therefore apply Theorem 7.1 to each $M_{i}$
to deduce there is a finite set of $(\Lambda,\sigma,\beta)$-smooth models
$\mathcal{S}$, and a constant $K$, and radii $r_{i}\in(4/10,1/2)$, so that
each $M_{i}\cap(B_{r_{i}},g_{\mathrm{eucl}})$ admits a
$(\Lambda,\beta,\mathcal{S},K)$-strong-cone decomposition. From (128), we get
that each $M_{i}\cap(B_{1},g_{\mathrm{eucl}})$ admits a
$(\Lambda,\beta,\mathcal{S},K+1)$-strong-cone decomposition. Ensuring
$\beta(\Lambda,\sigma,\epsilon)$ is sufficiently small and passing to a
subsequence, from Theorem 8.1 we can find a cone
$\hat{\mathbf{C}}\in\mathcal{C}_{\Lambda}$, and local bi-Lipschitz maps
$\hat{\phi}_{i}:B_{1}\to B_{1}$, satisfying Theorem 8.1(1)-(5) with
$M_{1},M_{i},\hat{\mathbf{C}}$ in place of $M_{v},M,\mathbf{C}_{v}$.

On the other hand, ensuring $\epsilon\leq\epsilon_{0}(\Lambda)/8$, we can
apply Lemma 8.5 to obtain $C^{2}$ diffeomorphisms $\psi_{i}:A_{\infty,1/4}\to
A_{\infty,1/4}$ satisfying

$\displaystyle\psi_{i}(M_{1}\cap A_{\infty,1/4})=M_{i}\cap
A_{\infty,1/4},\quad\psi_{i}|_{A_{1,1/4}}\text{ is a
$(\mathbf{C},\epsilon)$-map },$
$\displaystyle|D\psi_{i}|_{C^{0}(A_{R,1/2})}\leq c(R,M_{1})\text{ for every
$R\in(1/2,\infty)$}.$

If $m=1$, then Lemma 8.5 implies $|D\psi_{i}|\leq C$ on all of
$A_{\infty,1/4}$.

Theorem 8.1(5) and (128) imply that $d_{H}(\mathbf{C}\cap
B_{1},\hat{\mathbf{C}}\cap B_{1})\leq c(\Lambda)(\beta+\epsilon)$. Therefore,
arguing like we did in the proof of Theorem 2.4, provided
$\beta(\Lambda,\sigma),\epsilon(\Lambda,\sigma)$ are sufficiently small, we
can apply Lemma 8.9 to the maps $\hat{\phi}_{i}$, $\psi_{i}$, to obtain local
bi-Lipschitz $\phi_{i}:\mathbb{R}^{8}\to\mathbb{R}^{8}$ satisfying

$\displaystyle\phi_{i}(\overline{M_{1}})=\overline{M_{i}},\quad\phi_{i}(\mathrm{sing}M_{1})=\mathrm{sing}M_{i},\quad\phi_{i}|_{\mathbb{R}^{8}\setminus\mathrm{sing}M_{1}}\text{
is a $C^{2}$ diffeomorphism},$
$\displaystyle\mathrm{Lip}(\phi_{i}|_{B_{R}})\leq C(R)\text{ for every
$R\in\mathbb{R}$},$
$\displaystyle\phi_{i}|_{B_{1-10\sigma^{\prime}}}=\hat{\phi}_{i},\quad\phi_{i}|_{\mathbb{R}^{8}\setminus
B_{1-\sigma^{\prime}}}=\psi_{i},$

where $\sigma^{\prime}=\sigma/100$, and $C:\mathbb{R}\to\mathbb{R}$ is an
increasing function. This is a contradiction. ∎

##  11\. Appendix 1: a smallness estimate

In this section we prove an estimate which is a slight variation on Simon’s
asymptotic decay Theorem [Sim83a]. Following [Sim83a], we take $\Sigma$ a
smooth closed $n$-dimensional surface, and $\mathcal{E}$ a functional defined
on $C^{1}$ functions $u$ given by

$\mathcal{E}(u)=\int_{\Sigma}E(x,u,\nabla u),$

where $E$ satisfies the convexity and analyticity hypotheses of [Sim83a,
(1.2), (1.3)]. We write $\mathcal{M}(u)$ for the $-\mathrm{grad}\mathcal{E}$
in the $L^{2}$ sense (i.e. so that

$\left.\frac{d}{dt}\right|_{t=0}\mathcal{E}(u+t\zeta)=-\int_{\Sigma}\mathcal{M}(u)\zeta\quad\forall\zeta\in
C^{1}(\Sigma)$

holds), and $L$ for the linearization of $\mathcal{M}$ at $u=0$. As in
[Sim83a], we allow $u$ to take values in some vector bundle $V$ over $\Sigma$.

Let us recall some notation from [Sim83a]. Let $u$ be a $V$-valued function on
$\Sigma\times[0,T)$. Write $\nabla$ for the connection derivative on $V$, and
$\dot{u}\equiv\partial_{t}u$. Write $|u(t)|_{0}=|u(t,\cdot)|_{C^{0}(\Sigma)}$,
$||u(t)||=||u(t,\cdot)||_{L^{2}(\Sigma)}$, and

$\displaystyle|u|_{1}^{*}(t)$ $\displaystyle=|u(t)|_{0}+|\nabla
u(t)|_{0}+|\dot{u}|_{0},$ $\displaystyle|u|_{2}^{*}(t)$
$\displaystyle=|u|_{1}^{*}(t)+|\nabla^{2}u(t)|_{0}+|\nabla\dot{u}(t)|_{0}+|\ddot{u}(t)|_{0}.$

We consider here $C^{2}$ $V$-valued functions $u$ on $\Sigma\times[0,T)$
solving the PDE

(129)
$-\ddot{u}+m\dot{u}-\mathcal{M}(u)-\mathcal{R}_{1}(u)-\mathcal{R}_{2}(u)=f(x,t),$

where $m>0$, and:

  1. (1)

$f(x,t)$ is a $C^{2}$ function satisfying $|f|_{2}^{*}(t)\leq\delta
e^{-\epsilon t}$;

  2. (2)

$\mathcal{R}_{1}$ has the form

$\mathcal{R}_{1}(u)=(a_{1}\cdot\nabla^{2}u+a_{2})\dot{u}+a_{3}\cdot\nabla\dot{u}+a_{4}\ddot{u}$

where $a_{i}$ are $C^{2}$ functions of $(x,t,u,\nabla u,\dot{u})$ such that
$a_{i}(x,t,0,0,0)=0$ for $i=2,3,4$;

  3. (3)

$\mathcal{R}_{2}$ has the form

$\mathcal{R}_{2}(u)=b_{1}\cdot\nabla^{2}u+b_{2}\cdot\nabla\dot{u}+b_{3}\ddot{u}+b_{4}\cdot\nabla
u+b_{5}\dot{u}+b_{6}u,$

where $b_{i}$ are $C^{2}$ functions of $(x,t)$ such that
$|b_{i}|_{2}^{*}(t)\leq\delta e^{-\epsilon t}$.

The main content of our Theorem 11.1 is that one can replace Simon’s bounded
exponential growth condition [Sim83a, Definition 2] with a bounded linear
growth condition (Definition 11.0.1), at the expense of replacing a decay
estimate with a smallness estimate. Though the name suggests otherwise, our
linear growth condition is not a strictly weaker notion than Simon’s
exponential growth condition – only when $|u|$ is more-or-less larger than
$\delta$ are they comparable. Fortuitously, this is enough for our purposes.

The advantage is that the linear growth condition is effectively equivalent to
smallness of density drop of a minimal surface $M$, _without_ any assumptions
about the actual value of the density. For example, the linear growth
condition results directly from an assumption like
$\sup_{r}|\theta_{M}(0,r)-\theta_{\mathbf{C}}(0)|\leq\delta^{2}$. In contrast,
the exponential growth condition of [Sim83a] requires _additionally_ the
inequality $\sup_{r}\theta_{M}(0,r)\geq\theta_{\mathbf{C}}(0)$. This second
condition is too strong for our purposes, because in our annular cone regions
we only know $M$ is varifold close to $\mathbf{C}$, and $\theta_{M}(0,r)$ is
close to $\theta_{\mathbf{C}}(0)$. In particular, $\theta_{M}(0,r)$ may be
$>\theta_{\mathbf{C}}(0)$ at some radii, but $<\theta_{\mathbf{C}}(0)$ at
others.

On the other hand, the stronger assumption of Simon also gives a stronger
estimate. The exponential growth condition allows [Sim83a] to rule out
exponential growth of $u$ at any time, to deduce that $u$ exhibits only decay.
This also illustrates the asymmetry of the lower bound on $\theta_{M}(0,r)$:
if one instead assumed $\sup_{r}\theta_{M}(0,r)\leq\theta_{\mathbf{C}}(0)$,
one could show $u$ exhibits only growth (corresponding to uniqueness the
tangent cone of $M$ at infinity).

With our linear growth condition (130), we can only rule out exponential
growth (or decay) when $|u|>>\delta$, and so our Theorem 11.1 can only
conclude a smallness of $u$. $u$ may still exhibit both exponential growth and
decay. The linear growth condition is symmetric: smallness of $u$ propogates
both forward and backward in time. Of course, like [Sim83a], we can also give
a summability estimate (134) for $||\dot{u}(t)||$, as the decay/growth
behavior of $u$ is quantifiable.

######  Definition 11.0.1.

Given $K\geq 1$, $\delta>0$, we say a solution $u$ of (129) on
$\Sigma\times[0,T)$ has $(K,\delta)$-linear growth if

(130) $|u|^{*}_{2}(t)\leq\delta\max\\{1,|t-s|\\}+K|u|^{*}_{2}(s)$

for every $t,s,\in[0,T)$.

Our modification of [Sim83a, Theorem 1] is the following:

######  Theorem 11.1.

Take $T_{*}\geq T_{0}>0$, $K>0$. There are $\alpha(E),\epsilon(E)\in(0,1/2)$,
$\delta_{0}(\epsilon,K,T_{0},E,\mathcal{R}_{1})>0$ so that the following
holds. Let $\delta\leq\delta_{0}$, and let $u$ be a $C^{2}$ solution to (129)
on $\Sigma\times[0,T_{*})$ with $(K,\delta)$-linear growth, such that

(131) $\displaystyle|u|_{2}^{*}(t)\leq\delta\text{ for }t\in[0,T_{0}],$ (132)
$\displaystyle\mathcal{E}(u(t))\geq\mathcal{E}(0)-\delta\text{ for
}t\in[0,T_{*}),$

and Items 1-3 above hold for $f$, $\mathcal{R}_{1}$, $\mathcal{R}_{2}$ and all
$(x,t)\in\Sigma\times[0,T_{*})$. Then

(133) $|u|_{2}^{*}(t)<\delta^{\alpha}\quad\forall t\in[0,T_{*}),$

and

(134) $\int_{0}^{T_{*}}||\dot{u}(t)||dt\leq\delta^{\alpha/2}.$

######  Remark 11.2.

If $m<0$, then Theorem 11.1 continues to hold if instead of (132) we have
$\mathcal{E}(u(t))\leq\mathcal{E}(0)+\delta$ for all $t\in[0,T_{*})$.

###### Proof.

The proof is essentially the same as Simon’s. First note that by changing
variables $\tilde{t}=mt$ and replacing $E$ with $\frac{1}{m}E$, it suffices to
consider the case when $m=1$. We shall highlight: first, where Simon’s growth
condition is used, and how it can be replaced with our linear growth
assumption; and second, where $\mathcal{R}_{2}$ can be handled.

The strategy is to assume (after shrinking $T_{*}$ if necessary), that

(135) $|u|_{2}^{*}(t)<\delta^{\alpha}\quad\forall
t\in[0,T_{*}),\quad\limsup_{t\to T_{*}}|u|_{2}^{*}(t)=\delta^{\alpha}.$

and then use this to show that in fact

(136) $\limsup_{t\to T_{*}}|u|_{2}^{*}(t)<\delta^{\alpha},$

thereby establishing a contradiction. The choice of constants $\alpha$,
$\epsilon$, $\epsilon_{1}$, $\epsilon_{2}$, $\eta$, $R$, $\delta$ will satisfy
the same restrictions as in Simon, except we may take $R(\eta,\epsilon,E,K)$
larger than Simon, and $\delta(\epsilon,R,K,T_{0},E,\mathcal{R}_{1})$ smaller.

As noted in [Sim83a], if $\zeta$ is a solution to $\mathcal{M}(\zeta)=0$ with
$|\zeta|_{C^{2}(\Sigma)}\leq\delta^{2\alpha}$, then by differentiating (129)
the functions $u$, $w=\dot{u}$, $v=u-\zeta$ all solve equations of the form

(137)
$\displaystyle-\ddot{u}+\dot{u}-Lu=e_{1}\cdot\nabla^{2}u+e_{2}\cdot\nabla\dot{u}+e_{3}\ddot{u}+e_{4}\cdot\nabla
u+e_{5}\dot{u}+e_{6}u+f^{\prime},$

where $L$ is the linearization of $\mathcal{M}$ at $0$, and $e_{i}(t,x)$,
$f^{\prime}(t,x)$ are $C^{1}$ functions (different for each choice of $u,v,w$)
satisfying $|e_{j}|_{1}^{*}(t)\leq c(E,\mathcal{R}_{1})\delta^{\alpha}$,
$|f^{\prime}|_{1}^{*}(t)\leq 100\delta e^{-\epsilon t}$. Therefore, ensuring
$\delta(\epsilon,\eta,E,R,\mathcal{R}_{1})$ is small, we can apply the growth
theorem [Sim83a, Theorem 4] to each of $u$, $\dot{u}$, $u-\zeta$ to obtain
integers

$1\leq j_{1}\leq j_{2}\leq k-1,\quad 1\leq k_{1}\leq k_{2}\leq k-1,\quad 1\leq
l_{1}\leq l_{2}\leq k-1,$

so that [Sim83a, (5.9), (5.10), (5.11)] hold with $r_{1},r_{2}$ replaced by
$j_{1},j_{2}$ (in the case of $u$), $k_{1},k_{2}$ (in the case of $\dot{u}$),
and $l_{1},l_{2}$ (in the case of $u-\zeta$).

The first point where Simon uses his growth assumption is to show that
$j_{2}=l_{2}=k-1$. To show this using our linear growth assumption we argue as
follows. First note that provided $\delta(R,K)$ is sufficiently small, (135)
and (130) imply

(138) $|u|_{2}^{*}(t)\geq\delta^{\alpha}/(2K)\quad\text{ whenever }(k-3)R\leq
t<T_{*}.$

And then, if we had $j_{2}<k-1$, using (138), [Sim83a, (5.4), (5.11)], (135),
we would obtain the following contradiction for sufficiently large $R(E,K)$
and small $\delta(E,K)$:

(139) $\displaystyle\delta^{\alpha}/(2K)$
$\displaystyle\leq|u|_{2}^{*}((k-3/2)R)$ (140) $\displaystyle\leq
c(E)(\sup_{[(k-2)R,(k-1)R]}||u(t)||+\delta)$ (141) $\displaystyle\leq
c(E)(e^{-(\epsilon_{1}-\epsilon)R}\sup_{[(k-1)R,kR]}||u(t)||+\delta)$ (142)
$\displaystyle\leq c(E)(e^{-(\epsilon_{1}-\epsilon)R}\delta^{\alpha}+\delta).$

So $j_{2}=k-1$. A similar argument shows we must also have $l_{2}=k-1$, since
$|\zeta|_{C^{2}(\Sigma)}\leq\delta^{2\alpha}<<\delta^{\alpha}$.

The second (and only other) way in which exponential growth is used is to show
that bounds of $||u(t)||$ or $|u|_{2}^{*}(t)$ in terms of $\delta^{p}$, for
any particular $p\leq 1$, persist for nearby $t$. This also follows from our
linear growth assumption: if we have

$|u|_{2}^{*}(t)\leq
C\delta^{p}\quad\text{or}\quad\sup_{[jR,(j+1)R]}||u(t)||\leq C\delta^{p}$

for some $C>0$, $p\leq 1$, $t\in[0,T_{*})$ or $0\leq j<k-1$, then (130) and
[Sim83a, (5.4)] imply

$|u|_{2}^{*}(s)\leq
c(C,K,R)\delta^{p},\quad\text{or}\quad\sup_{[j^{\prime}R,(j^{\prime}+1)R^{\prime}]}||u(t)||\leq
c(C,K,R)\delta^{p}$

(respectively) for some constant $c(C,K,R)$, provided $|s-t|\leq 10R$,
$|j-j^{\prime}|\leq 6$, $0\leq j^{\prime}\leq k-1$, and $\delta(R,K,p)$ is
sufficiently small.

The second remainder term $\mathcal{R}_{2}$ can be dealt with in the same
fashion as $f$. Specifically, we have by [Sim83a, (6.26)] and our bound
$|u|_{2}^{*}\leq\delta^{\alpha}$ that

$||\mathcal{R}_{2}(t)||+||f(t)||\leq
c(E)\delta^{1/2}||\dot{u}(t)||\quad\forall t\in(k_{1}R,(k_{2}-1)R),$

which is used in applying [Sim83a, Lemma 1] to obtain [Sim83a, (6.34)]; and by
[Sim83a, (6.46)],

$||\mathcal{R}_{2}((k_{2}+2)R)||+||f((k_{2}+2)R)||\leq
c(E)\delta^{1/2}\sup_{[(k_{2}+1)R,(k_{2}+3)R]}||\dot{u}(t)||,$

which is used in the estimate for $||v(t)||$ on [Sim83a, Page 557]. The rest
of the proof of estimate (133) proceeds as in [Sim83a].

To prove (134), first note that we have by (133) the bound $||\dot{u}(t)||\leq
c(\Sigma)\delta^{\alpha}$ for all $t$. Now by [Sim83a, (6.31), (6.34)] we get

$\int_{0}^{(k_{2}-1)R}||\dot{u}||dt\leq
c(\epsilon)\delta^{1/2}+c(E)\delta^{\theta}+c(E)\delta^{\alpha}.$

For $k_{2}-1\leq j\leq k-2$, note that from [Sim83a, (6.41)] we get
$\sup_{[jR,(j+1)R]}||\dot{u}(t)||\leq
c(\Sigma)e^{-(\epsilon_{1}-\epsilon)(j-k+1)R}\delta^{\alpha}$, and therefore

$\int_{(k_{2}-1)R}^{T_{*}}||\dot{u}||dt\leq
c(E,R)\delta^{\alpha}\sum_{j=0}^{k-k_{2}}e^{-(\epsilon_{1}-\epsilon)j}\leq
c(E,R,\epsilon)\delta^{\alpha}.$

Ensuring $\delta(E,R,\epsilon)$ is sufficiently small, we obtain (134). ∎

##  12\. Appendix 2: a function

In constructing our parameterization we used the following $1$-dimensional
Lemma.

######  Lemma 12.1.

Given $\epsilon\in(0,1]$, $k\in\mathbb{N}$, let
$\mathcal{A}_{\epsilon,k}=\\{(a_{1},\ldots,a_{k})\in\mathbb{R}^{k}:-\epsilon<a_{1}<a_{2}<\ldots<a_{k}<\epsilon\\}$.
Then there is a smooth function $f\equiv
f_{\\{a_{i}\\},\\{b_{i}\\}}(t):\mathcal{A}_{\epsilon,k}\times\mathcal{A}_{\epsilon,k}\times\mathbb{R}\to\mathbb{R}$
satisfying:

$f(a_{i})=b_{i},\quad f|_{\\{|t|\geq
2\epsilon\\}}=t,\quad\frac{9}{10}\underline{M}\leq\partial_{t}f\leq
11\overline{M}$

and

$|\partial_{a_{j}}f|\leq c\overline{M},\quad|\partial_{b_{j}}f|\leq c,$

where

$\overline{M}=\max\left\\{\max_{i}\frac{|b_{i+1}-b_{i}|}{|a_{i+1}-a_{i}|},3\right\\},\quad\underline{M}=\min\left\\{\min_{i}\frac{|b_{i+1}-b_{i}|}{|a_{i+1}-a_{i}|},\frac{1}{3}\right\\},$

and $c$ is an absolute constant.

######  Remark 12.2.

We highlight two basic facts about $f$. First, we have
$f_{\\{a_{i}\\},\\{a_{i}\\}}(t)=t$. Second, if $\tilde{a}_{i}=-a_{k-i}$ and
$\tilde{b}_{i}=-b_{k-i}$, and
$\tilde{f}=f_{\\{\tilde{a}_{i}\\},\\{\tilde{b}_{i}\\}}$, then we have
$f_{\\{a_{i}\\},\\{b_{i}\\}}(t)=-\tilde{f}(-t)$.

###### Proof.

Fix $\underline{m},\overline{m}:\mathbb{R}^{2}\to\mathbb{R}$ to be
$1$-homogenous functions, smooth away from $0$, such that

(143)
$\displaystyle\min\\{s_{1},s_{2}\\}\leq\underline{m}(s_{1},s_{2})\leq(101/100)\min\\{s_{1},s_{2}\\},\quad\underline{m}(s,s)=s,$
(144) $\displaystyle\quad\overline{m}(s_{1},s_{2})\geq\max\\{s_{1},s_{2}\\}.$

Fix $\eta:\mathbb{R}\to\mathbb{R}$ a smooth function satisfying

$\eta|_{(-\infty,-1]\cup[1,\infty)}=0,\quad\eta|_{[-1/4,1/4]}=1,\quad
t\eta^{\prime}(t)\leq 0,\quad|\eta^{\prime}|\leq 10.$

Define $m:\mathbb{R}^{3}\to\mathbb{R}$ by setting

$m(s_{1},s_{2},t)=\left\\{\begin{array}[]{l
l}s_{1}+\eta(t)(\underline{m}(s_{1},s_{2})-s_{1})&t<0\\\
s_{2}+\eta(t)(\underline{m}(s_{1},s_{2})-s_{2})&t\geq 0\end{array}\right..$

Trivially $m$ is smooth on $\\{s_{1},s_{2}>0\\}\times\mathbb{R}$, and in this
domain $m$ satisfies

$m+|\partial_{t}m|\leq 20\max\\{s_{1},s_{2}\\},\quad|\partial_{s_{i}}m|\leq
c,$

where $c=\max|D\underline{m}|$ is an absolute constant. $m$ is our transition
gradient.

Define $\hat{f}(s_{1},s_{2},t)=m(s_{1},s_{2},t)t$. Trivially $\hat{f}$ is
smooth on $\\{s_{1},s_{2}>0\\}\times\mathbb{R}$, and $\hat{f}$ satisfies

$\hat{f}|_{(-\infty,-1]}=s_{1}t,\quad\hat{f}|_{[1,\infty)}=s_{2}t,\quad\hat{f}(s_{1},s_{2},0)=0.$

A straightforward computation shows that

$(9/10)\min\\{s_{1},s_{2}\\}\leq\partial_{t}\hat{f}\leq
11\max\\{s_{1},s_{2}\\},\quad|\partial_{s_{i}}\hat{f}|\leq c|t|,$

with $c$ an absolute constant. $\hat{f}$ is our model transition function,
between two linear functions of different positive gradients.

Take $\\{a_{i}\\}_{i=1}^{k},\\{b_{i}\\}_{i=1}^{k}\in\mathcal{A}_{\epsilon,k}$.
Let us define $a_{0}=b_{0}=-\frac{3}{2}\epsilon$,
$a_{k+1}=b_{k+1}=\frac{3}{2}\epsilon$, and

(145) $\displaystyle r_{i}$
$\displaystyle=\overline{m}\left(\frac{2}{a_{i+1}-a_{i}},\frac{2}{a_{i}-a_{i-1}}\right)\quad
i=1,\ldots,k,$ (146) $\displaystyle m_{i}$
$\displaystyle=\frac{b_{i+1}-b_{i}}{a_{i+1}-a_{i}},\quad i=0,\ldots,k.$

Now we let

$f(\\{a_{i}\\},\\{b_{i}\\},t)=\left\\{\begin{array}[]{l
l}\frac{\epsilon}{4}\hat{f}(1,m_{0},\frac{4}{\epsilon}(t+\frac{3}{2}\epsilon))-\frac{3}{2}\epsilon&t\leq
a_{1}-\frac{1}{r_{1}}\\\
\frac{1}{r_{i}}\hat{f}(m_{i-1},m_{i},r_{i}(t-a_{i})+b_{i}&a_{i}-\frac{1}{r_{i}}\leq
t\leq a_{i}+\frac{1}{r_{i}}\\\ m_{i}(t-a_{i})+b_{i}&a_{i}+\frac{1}{r_{i}}\leq
t\leq a_{i+1}-\frac{1}{r_{i+1}}\\\
\frac{\epsilon}{4}\hat{f}(m_{k},1,\frac{4}{\epsilon}(t-\frac{3}{2}\epsilon))+\frac{3}{2}\epsilon&t\geq
a_{k}+\frac{1}{r_{k}}\end{array}\right..$

Then one can readily check $f$ is smooth, well-defined, and satisfies

$(9/10)\min_{i}\\{m_{i}\\}\leq\partial_{t}f\leq
11\max_{i}\\{m_{i}\\},\quad|\partial_{a_{j}}f|\leq
c\max_{i}\\{m_{i}\\},\quad|\partial_{b_{j}}f|\leq c,$

for any $j=1,\ldots,k$, and for $c$ being an absolute constant. This proves
the Lemma. ∎

##  13\. Appendix 3: quadratic cones

In this section we prove Theorem 2.8. We shall work in $\mathbb{R}^{n+1}$, for
$n\geq 7$. We will want to use a slight variant of Theorem 6.3, the proof
being virtually verbatim (in fact easier). Let us state that first.

######  Theorem 13.1.

Let $\mathbf{C}^{n}$ be a smooth (away from $0$), stationary cone in
$\mathbb{R}^{n+k}$. Given any $\epsilon>0$, there are constants
$\delta(\mathbf{C},k,\epsilon)$, $\beta(\mathbf{C},k,\epsilon)$ so that the
following occurs. Let $\rho\geq 0$, $g$ be a $C^{3}$ metric on $B_{1}$
satisfying $|g-g_{\mathrm{eucl}}|_{C^{3}(B_{1})}\leq\beta$, and let $V$ be a
stationary integral $n$-varifold in $(B_{1},g)$. Suppose there is a $C^{2}$
function $u:\mathbf{C}\cap A_{1,1/2}\to\mathbf{C}^{\perp}$ so that

(147) $\displaystyle\mathrm{spt}V\cap
A_{1,1/2}=G_{\mathbf{C}}(u),\quad|u|_{C^{2}}\leq\delta$ (148)
$\displaystyle\theta_{V}(0,1)\leq\theta_{\mathbf{C}}(0)+\beta,\quad\theta_{V}(0,\rho/2)\geq\theta_{\mathbf{C}}(0)-\beta.$

Then $u$ can be extended to a $C^{2}$ function on $\mathbf{C}\cap A_{1,\rho}$,
so that

$\mathrm{spt}V\cap A_{1,\rho}=G_{\mathbf{C}}(u)\cap
A_{1,\rho},\quad|x|^{-1}|u|+|\nabla u|+|x||\nabla^{2}u|\leq\epsilon.$

###### Proof.

Same as Theorem 6.3, except we use Lemma 13.2 in place of Lemma 6.2. ∎

######  Lemma 13.2.

Let $\mathbf{C}^{n}$ be a smooth (away from $0$), stationary cone in
$\mathbb{R}^{n+k}$. Given $\epsilon>0$, there is a
$\beta(\mathbf{C},k,\epsilon)$ so that the following occurs. Let $g$ be a
$C^{3}$ metric on $B_{1}$ satisfying
$|g-g_{\mathrm{eucl}}|_{C^{2}(B_{1})}\leq\beta$, and let $V$ be a stationary
integral $n$-varifold in $(B_{1},g)$ satisfying

(149) $\displaystyle\mathrm{spt}V\cap
A_{1,1/2}=G_{\mathbf{C}}(u),\quad|u|_{C^{2}}\leq\beta,$ (150)
$\displaystyle\theta_{V}(0,1)\leq\theta_{\mathbf{C}}(0)+\beta,\quad\theta_{V}(0,1/8)\geq\theta_{\mathbf{C}}(0)-\beta.$

Then

(151) $\mathrm{spt}V\cap A_{1,1/4}=G_{\mathbf{C}}(u)\cap
A_{1,1/4},\quad|u|_{C^{2}}\leq\epsilon.$

###### Proof.

Proof by contradiction: otherwise, there are $\beta_{i}\to 0$, metrics $g_{i}$
such that $|g_{i}-g_{\mathrm{eucl}}|_{C^{3}(B_{1})}\leq\beta_{i}$, and $V_{i}$
being stationary integral varifolds in $(B_{1},g_{i})$ such that (149), (150)
hold with $\beta_{i}$ in place of $\beta$, but (151) fails. Passing to a
subsequence, we can assume $V_{i}\to V$ as varifolds for some stationary
integral varifold $V$ in $(B_{1},g_{\mathrm{eucl}})$. Since
$\theta_{V}(0,1)\leq\theta_{\mathbf{C}}(0)$ and $\lim_{r\downarrow
1/8}\theta_{V}(0,r)\geq\theta_{\mathbf{C}}(0)$, monotonicity implies $V$ is
dilation-invariant in $A_{1,1/8}$. On the other hand, by (149) we can assume
that $V_{i}\to[\mathbf{C}]$ in $C^{2}$ with multiplicity-one in $A_{1,1/2}$.
So we must have $V\llcorner A_{1,1/8}=[\mathbf{C}]\llcorner A_{1,1/8}$.
Allard’s theorem implies (151) holds for $i>>1$, which is a contradiction. ∎

We could use the cone regions from Section 6, but to avoid clashes of notation
and because everything in this proof is vastly easier, let us use instead the
following simpler notion of cone regions.

######  Definition 13.2.1.

Let $V$ be a varifold in $B_{R}(a)$. We say $V\llcorner A_{R,\rho}(a)$ is a
$(\mathbf{C},\beta,\tau)$-weak-cone$*$ region if for every
$r\in(\rho/4,(1-10^{-2})R]$, there is an $a_{r}\in B_{R}(a)$ with
$B_{r}(a_{r})\subset B_{R}(a)$, so that

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

and if $s\in(\rho/4,(1-10^{-2})R]\cap[r/2,2r]$ then

$|a_{r}-a_{s}|\leq\tau\max\\{r,s\\}.$

We say $V\llcorner A_{R,\rho}(a)$ is a $(\mathbf{C},\beta)$-cone$*$ region if
for every $r\in(\rho/4,R]$ we have

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

If we think of $\sigma=10^{-2}$, $m=1$, and replace $c(\Lambda,m)$ with
$c(\mathbf{C})$, then Lemma 6.1 and its proof continue to hold for cone$*$
regions. We are now prepared to prove Theorem 2.8.

###### Proof of Theorem 2.8.

We prove Theorem 2.8 by contradiction. We will choose our constants as we go
along, and a posteriori they can all be fixed, but the reader should think of
them like:

$\beta^{\prime\prime}<<\tau<<\beta^{\prime}<<\epsilon^{\prime}<<\epsilon<1.$

Suppose the theorem fails: then there is a sequence of numbers $\delta_{i}\to
0$, metrics $g_{i}$ such that
$|g_{i}-g_{\mathrm{eucl}}|_{C^{3}(B_{1})}\leq\delta_{i}$, and stationary
integral varifolds $V_{i}$ in $(B_{1},g_{i})$, so that (7) holds with
$\delta_{i}$, $g_{i}$, $V_{i}$ in place of $\delta$, $g$, $V$, but (9) fails
for any $a,\lambda,q$ satisfying (8). Passing to a subsequence, standard
compactness for integral varifolds implies we can find a stationary integral
varifold $V$ in $(B_{1},g_{\mathrm{eucl}})$, so that $V_{i}\to V$ as
varifolds. By our hypotheses and the constancy theorem we must have
$V=[\mathbf{C}]$, and so by Allard’s theorem there is no loss in assuming
$V_{i}\to[\mathbf{C}]$ as varifolds in $B_{1}$ and in $C^{2}$ on compact
subsets of $B_{1}\setminus\\{0\\}$.

Let $\rho_{i}$ be the least number such that $V_{i}\llcorner
A_{1,\rho_{i}}(0)$ is a $(\mathbf{C},\beta^{\prime\prime},\tau)$-weak-cone$*$
region. Let $a_{i}=a_{\rho_{i}}(V_{i})$ as in Definition 13.2.1, and then by
our convergence $V_{i}\to[\mathbf{C}]$ we have $a_{i}\to 0$, $\rho_{i}\to 0$.
By Lemma 6.1, ensuring $\beta^{\prime\prime}(\mathbf{C},\beta^{\prime})$,
$\tau(\mathbf{C},\beta^{\prime})$ are sufficiently small, each $V_{i}\llcorner
A_{9/10,\rho_{i}}$ is a $(\mathbf{C},\beta^{\prime})$-cone$*$ region, and
hence by our $C^{2}$ convergence and by Theorem 13.1, ensuring
$\beta^{\prime}(\mathbf{C},\epsilon^{\prime})$ is small and $i>>1$, we can
find $C^{2}$ functions $u_{i}:\mathbf{C}\cap
A_{9/10,\rho_{i}/2}\to\mathbf{C}^{\perp}$ so that

(152) $\mathrm{spt}V_{i}\cap
A_{9/10,\rho_{i}/2}(a_{i})=a_{i}+G_{\mathbf{C}}(u_{i})\cap
A_{9/10,\rho_{i}/2},\quad|x|^{-1}|u_{i}|+|\nabla
u_{i}|+|x||\nabla^{2}u_{i}|\leq\epsilon^{\prime}\leq\epsilon.$

If $\rho_{i}=0$ for infinitely-many $i$ then by (152) we obtain a
contradiction for $i$ large. Suppose now every $\rho_{i}>0$. Define the
rescaled varifolds $V_{i}^{\prime}=(\eta_{a_{i},\rho_{i}})_{\sharp}V_{i}$, and
metrics $g_{i}^{\prime}=g_{i}\circ\eta_{a_{i},\rho_{i}}^{-1}$. Then for every
$R>2$ and $i>>1$, we have

(153) $1=\inf\\{\rho:V_{i}^{\prime}\llcorner A_{R,\rho}(0)\text{ is a
$(\mathbf{C},\beta^{\prime\prime},\tau)$-weak-cone$*$ region}\\},$

and $V_{i}^{\prime}$ is stationary in $(B_{R},g_{i}^{\prime})$, and
$g_{i}^{\prime}\to g_{\mathrm{eucl}}$ in $C^{2}$ on compact subsets of
$\mathbb{R}^{n+1}$.

Monotonicity (18) implies that
$\theta_{V_{i}^{\prime}}(0,R)\leq\theta_{\mathbf{C}}(0)+o(1)$, and so we can
pass to a subsequence and obtain a stationary integral varifold $V^{\prime}$
in $(\mathbb{R}^{n+1},g_{\mathrm{eucl}})$ satisying
$\theta_{V^{\prime}}(0,\infty)\leq\theta_{\mathbf{C}}(0)$, so that
$V_{i}^{\prime}\to V^{\prime}$ as varifolds. From (152), Arzela-Ascoli, and
standard elliptic estimates, provided $\epsilon^{\prime}(\mathbf{C})$ is
small, we can assume that

(154) $\mathrm{spt}V^{\prime}\cap
A_{\infty,1/2}=G_{\mathbf{C}}(u^{\prime})\cap
A_{\infty,1/2},\quad|x|^{-1}|u^{\prime}|+|\nabla
u^{\prime}|+|x||\nabla^{2}u^{\prime}|\leq\epsilon^{\prime},$

and $V_{i}^{\prime}\to V^{\prime}$ in $C^{2}$ on compact subsets of
$A_{\infty,1/2}$. We can also assume that $V^{\prime}\llcorner A_{\infty,1}$
is a $(\mathbf{C},\beta^{\prime})$-cone$*$ region.

Together the previous two sentences imply that if $V^{\prime\prime}$ is a
tangent cone of $V^{\prime}$ at infinity, then
$d_{H}(\mathrm{spt}V^{\prime\prime}\cap B_{1},\mathbf{C}\cap
B_{1})\leq\epsilon^{\prime}$, and
$|\theta_{V^{\prime\prime}}(0)-\theta_{\mathbf{C}}(0)|\leq\beta^{\prime}$.
Ensuring $\epsilon^{\prime}(\mathbf{C})$, $\beta^{\prime}(\mathbf{C})$ is
small, we deduce by integrability of $\mathbf{C}$ that
$V^{\prime\prime}=[q(\mathbf{C})]$ for some rotation $q\in SO(n+1)$ satisfying
$|q-Id|\leq c(\mathbf{C})\epsilon^{\prime}$. [SS86] or [ES19, Corollary 3.7]
then implies that $V^{\prime}=[b+q(S_{\lambda})]$ for some leaf $S_{\lambda}$
of the Hardt-Simon foliation, where we allow $S_{0}=\mathbf{C}$. We aim to
show $\lambda\neq 0$.

I claim there is an $\eta(\mathbf{C},\tau)$ so that if $a\in B_{1}$,
$r\in[1/2,4]$, and

$|\theta_{\mathbf{C}}(a,r)-\theta_{\mathbf{C}}(0)|\leq\eta,$

then $|a|\leq 2^{-10}\tau$. Suppose otherwise: there are sequences
$\eta_{i}\to 0$, $a_{i}\in B_{2}$, $r_{i}\in[1/2,4]$ so that
$|\theta_{\mathbf{C}}(a_{i},r_{i})-\theta_{\mathbf{C}}(0)|\leq\eta_{i}$, but
$|a_{i}|\geq 2^{-10}\tau$. Passing to a subsequence, we can assume $a_{i}\to
a\in\overline{B_{1}}$, $r_{i}\to r\in[1/2,2]$, and so by monotonicity we get
$\theta_{\mathbf{C}}(a,2r)=\theta_{\mathbf{C}}(0)$, and hence $\mathbf{C}\cap
A_{\infty,2r}(a)=(a+\mathbf{C}^{\prime})\cap A_{\infty,2r}(a)$ for some cone
$\mathbf{C}^{\prime}$. Since $\mathbf{C}$ has no symmetries, we must have
$\mathbf{C}^{\prime}=\mathbf{C}$ and $a=0$. This is a contradiction.

Let us ensure $\beta^{\prime\prime}\leq\eta(\mathbf{C},\tau)$, and $\tau\leq
2^{-4}$. Assume that $V^{\prime}=[b+q(\mathbf{C})]$. By (154), $V^{\prime}$ is
regular outside $\overline{B_{1/2}}$, so we must have
$b\in\overline{B_{1/2}}$. I next claim that if
$a_{i,r}=a_{r}(V_{i}^{\prime})$, then for all $i>>1$ and $r\in[1/2,2]$ we must
have

$|a_{i,r}-b|\leq 2^{-9}\tau.$

Note that by construction we have $a_{i,1}=0$, and (hence) $a_{i,r}\in
B_{1/4}$. If my second claim failed, then we could find a sequence
$r_{i}\in[1/2,2]$ so that $|a_{i,r_{i}}|\geq 2^{-8}\tau$. Passing to a
subsequence, we can assume $a_{i,r_{i}}\to a\in\overline{B_{1/4}}$ and
$r_{i}\to r\in[1/2,2]$, and hence
$|\theta_{b+q(\mathbf{C})}(a,s)-\theta_{\mathbf{C}}(0)|\leq\beta^{\prime\prime}$
for all $s>r$. From my first claim (applied in the ball $B_{1}(b)$), we deduce
that $|a-b|\leq 2^{-10}\tau$. This proves my second claim.

Varifold convergence $V_{i}^{\prime}\to V^{\prime}$ implies
$|\theta_{V_{i}^{\prime}}(b,r)-\theta_{\mathbf{C}}(0)|\leq\beta^{\prime\prime}$
for $i>>1$ and $r\in[1/16,2]$, and therefore combined with the previous
paragraph we get that $V_{i}^{\prime}\llcorner B_{4}(0)\setminus B_{1/2}$ is a
$(\mathbf{C},\beta^{\prime\prime},\tau)$-cone$*$ region for $i$ large. This
contradicts (153), so we must have $\lambda\neq 0$.

Now $V^{\prime}=[b+q(S_{\lambda})]$ is smooth, multiplicity-one, and hence by
Allard’s theorem $V_{i}^{\prime}\to V^{\prime}$ in $C^{2}$ with multiplicity-
one. Since $|b|\leq 1$ and $\mathrm{spt}V^{\prime\prime}\cap
B_{1}\neq\emptyset$, we have an upper bound
$|\lambda|\leq\lambda_{0}(\mathbf{C})$, and hence there is an
$R(\mathbf{C},\epsilon^{\prime})$ so that

(155) $(b+q(S_{\lambda}))\cap
A_{\infty,R}=G_{q(\mathbf{C})}(v^{\prime}),\quad|x|^{-1}|v^{\prime}|+|\nabla
v^{\prime}|+|x||\nabla^{2}v^{\prime}|\leq\epsilon^{\prime}.$

For $i>>1$, by $C^{2}$ convergence we can find
$u_{i}^{\prime}:(b+q(S_{\lambda}))\cap B_{2R}\to S_{\lambda}^{\perp}$ so that

(156) $\mathrm{spt}V_{i}^{\prime}\cap
B_{2R}=\mathrm{graph}_{b+q(S_{\lambda})}(u_{i}^{\prime}),\quad|x|^{-1}|u_{i}^{\prime}|+|\nabla
u_{i}^{\prime}|+|x||\nabla^{2}u_{i}^{\prime}|\leq\epsilon^{\prime}.$

If we let
$b_{i}+q(S_{\lambda_{i}})=\eta_{a_{i},\rho_{i}}^{-1}(b+q(S_{\lambda}))$, and
ensure $\epsilon^{\prime}(\mathbf{C},\epsilon)$ is sufficiently small, then
(152), (155), (156) imply (8), (9) hold with $b_{i},q,\lambda_{i}$ in place of
$a_{i},q,\lambda_{i}$. This is a contradiction, and finishes the proof of
Theorem 2.8 for quadratic cones.

For general strictly-minimizing and strictly-stable
$\mathbf{C}^{n}\subset\mathbb{R}^{n+1}$, and $\mathrm{spt}V$ lying to one side
of $\mathbf{C}^{n}$, the proof proceeds in a similar but even simpler fashion.
Take $V_{i}$, $g_{i}$ the counter-example sequence as before, and then let
$\rho_{i}$ be the least radius so that $V_{i}\llcorner A_{9/10,\rho_{i}}(0)$
is a $(\mathbf{C},\beta^{\prime})$-cone$*$ region. Our convergence
$V_{i}\to[\mathbf{C}]$ implies $\rho_{i}\to 0$, and ensuring
$\beta^{\prime}(\mathbf{C},\epsilon)$ is small, we can apply Theorem 13.1 to
deduce (152) holds with $a_{i}=0$.

Let $V_{i}^{\prime}=(\eta_{0,\rho_{i}})_{\sharp}V_{i}$, and
$g_{i}^{\prime}=g\circ\eta_{0,\rho_{i}}^{-1}$. Then for every $R>2$ and $i>>1$
we have

(157) $1=\inf\\{\rho:V_{i}^{\prime}\llcorner A_{R,\rho}(0)\text{ is a
$(\mathbf{C},\beta^{\prime})$-cone$*$ region}\\},$

$V_{i}^{\prime}$ is stationary in $(B_{R}(0),g_{i}^{\prime})$,
$\mathrm{spt}V_{i}^{\prime}$ lies to one-side of $\mathbf{C}^{n}$, and
$g_{i}^{\prime}\to g_{\mathrm{eucl}}$ in $C^{2}(B_{R}(0))$.

As before, by monotonicity and standard compactness we can pass to a
subsequence and assume that $V_{i}^{\prime}\to V^{\prime}$, for $V^{\prime}$ a
stationary integral varifold in $(\mathbb{R}^{n+1},g_{\mathrm{eucl}})$ such
that $\theta_{V^{\prime}}(0,\infty)\leq\theta_{\mathbf{C}}(0)$, and
$\mathrm{spt}V^{\prime}$ lies to one side of $\mathbf{C}^{n}$. [Sim21a, Lemma
7.6] implies that $V^{\prime}=[S_{\lambda}]$ for some $\lambda$. If we had
$\lambda=0$, then we would have
$\theta_{V_{i}^{\prime}}(0,r)\to\theta_{V^{\prime}}(0,r)=\theta_{\mathbf{C}}(0)$
for every $r>0$, and hence by monotonicity $V_{i}^{\prime}\llcorner
A_{4,1/2}(0)$ would be a $(\mathbf{C},\beta^{\prime})$-cone$*$ region for
$i>>1$, contradicting (157). So we must have $\lambda\neq 0$, and then the
proof proceed in the same way as above. ∎

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