# Regularity of free boundary minimal surfaces in locally polyhedral domains

Nick Edelen and Chao Li  Nick Edelen:
University of Notre Dame, 255 Hurley Bldg, Notre Dame, IN 46556, USA
E-mail address<EMAIL_ADDRESS>

Chao Li:
Princeton University, Fine Hall, 304 Washington Road, Princeton, NJ 08540, USA
E-mail address<EMAIL_ADDRESS>
###### Abstract.

We prove an Allard-type regularity theorem for free-boundary minimal surfaces
in Lipschitz domains locally modelled on convex polyhedra. We show that if
such a minimal surface is sufficiently close to an appropriate free-boundary
plane, then the surface is $C^{1,\alpha}$ graphical over this plane. We apply
our theorem to prove partial regularity results for free-boundary minimizing
hypersurfaces, and relative isoperimetric regions.

###### Contents

  1. 1 Introduction
  2. 2 Notation and preliminaries
    1. 2.1 Notation
    2. 2.2 Polyhedral cone domains
    3. 2.6 Curved polyhedral cones
    4. 2.9 Varifolds
    5. 2.10 Excess
    6. 2.11 Changing cone type
  3. 3 Main theorem
    1. 3.8 Outline of proof
  4. 4 First variation, mass control
  5. 5 Eigenvalues and the linearized operator
  6. 6 $L^{2}$ excess decay
  7. 7 Regularity
  8. 8 Higher codimension and ambient manifolds
  9. 9 Minimizers
  10. 10 Appendix: Tangential first variation
    1. 10.1 Free-boundary
    2. 10.3 Prescribed boundary
    3. 10.5 Higher codimension boundary
  11. 11 Appendix: First variation and Sobolev inequalities
  12. 12 Appendix: Fourier expansion in cones

##  1\. Introduction

We are interested in the regularity of free-boundary minimal hypersurfaces $M$
inside piecewise-smooth domains $\Omega$. These surfaces $M$ arise
variationally as critical points of area or capillary type functions among
surfaces in $\Omega$ whose boundaries lie in $\partial\Omega$ but are
otherwise free to vary. The existence and regularity of free-boundary minimal
surfaces has been extensively studied by Courant [6], Lewy [26, 25],
Hildebrandt [18], Nitsche [17, 31], Gruter [13] and Jost [14], Taylor [39,
38], Struwe [37], among others.

When $\partial\Omega$ is at least $C^{2}$, Gruter-Jost [14] proved an Allard-
type regularity theorem which says that if $M$ is sufficiently varifold close
to a free-boundary plane (or, equivalently in this case, when the density
ratio of $M$ near a boundary point is sufficiently close to $1/2$), then
nearby $M$ is a $C^{1,\alpha}$ graph over this plane. Gruter [13] used this
regularity theorem to prove that the (optimal) dimension of the singular set
at the free-boundary for area-minimizing hypersurfaces satisfies the same
codimension $7$ bound as in the interior (contrast this with Hardt-Simon’s
[16] result that showed every area-minimizing hypersurface is entirely regular
near its Dirichlet boundary).

In this paper we prove an Allard-type regularity for $M$ when $\Omega$ is
locally modeled on any convex polyhedral cone, and hence is only piecewise
$C^{2}$. The archtype of such domain is a $C^{2}$ perturbation of a convex
wedge (i.e. intersection of two half-spaces). Existence and regularity of
free-boundary minimal $M$ in non-smooth $\Omega$ has a history dating back to
the 19th century, when Gergonne [11] and Schwartz [32] formulated and solved
the question of determining minimal surfaces with partially free boundary in a
standard cube in $\mathbb{R}^{3}$. Since then, there has been a rich history
of investigation on free boundary minimal surfaces in various geometric and
physical scenarios, where polyhedral domains naturally arise. For instance,
when $\Omega$ is a wedge region in $\mathbb{R}^{3}$, with opening angle
$\theta_{0}$, a range of geometric and regularity properties of free boundary
minimal surfaces were studied in [19, 20, 21, 22]. See also [17], where
$\theta_{0}=2\pi$.

In these past results, as well as in [23, 15], it was observed that boundary
regularity of minimal surfaces depends on the local structure of $\Omega$. For
instance, if $\Omega$ is a wedge in $\mathbb{R}^{3}$ as above, and
$\theta_{0}=2\pi$, then an area minimizing surface $M$ may have branch points
at its free boundary (see [17]). More generally, as pointed in [20], when
$\theta_{0}\in(\pi,2\pi)$, an area-minimizing surface in $\Omega$ may contain
an interval on the edge $\\{0\\}\times\mathbb{R}$, and thus fail to be a
regular surface meeting $\partial\Omega$ orthogonally (called the edge-
creeping phenomenon). For a beautuful local description of two-dimensional
minimal surfaces in wedges using the Weierstrass representation, see [3].

In polyhedral domains, similar regularity questions also naturally appear in
geometric problems with other type of boundary conditions, including capillary
surfaces [34] [38] as well as general soap bubbles [12]. These surfaces are
crucial in Gromov’s geometric comparison principle for scalar curvature proved
by the second named author [28, 27].

Simon [34] implemented tools of geometric measure theory in the investigation
of capillary surfaces in domains with corners. For a free boundary minimal
surface, he noted that local convexity of the model domain is a sufficient
condition for the existence of a nontrivial tangent cone at the corner. On the
other hand, cusp type singularities may occur at the corner if local convexity
is violated (see [34][4][5] and the references therein).

Our main result is a local Allard-type regularity theorem, which says loosely
that if $\Omega\cap B_{1}$ is a sufficiently small $C^{2}$ perturbation of a
convex polyhedral cone ${\Omega^{(0)}}$, and if $M\llcorner B_{1}$ is
sufficiently varifold close to an appropriate free-boundary plane in
${\Omega^{(0)}}$, then $\mathrm{spt}M\cap B_{1/2}$ is a $C^{1,\alpha}$
perturbation of this plane.

######  Theorem 1.1.

Let $\Omega^{n+1}=\Omega_{0}\times\mathbb{R}$ be a polyhedral cone domain: a
dilation-invariant intersection of finitely-many closed half-spaces, with non-
empty interior. Let $F_{i}:B_{1}\to B_{1}$ be a sequence of $C^{2}$
diffeomorphisms which limit to $\mathrm{id}$ in $C^{2}(B_{1})$. Let $M_{i}$ be
a sequence of integral varifolds in $B_{1}$ which are stationary free-boundary
in $F_{i}(\Omega)$ (or have bounded mean curvature tending to $0$), such that
$M_{i}\to[\Omega_{0}\times\\{0\\}]$ as varifolds in $B_{1}$.

Then for $i\gg 1$, we can write $F_{i}^{-1}(\mathrm{spt}M)\cap B_{1/2}$ as
$C^{1,\alpha}$ graphs over $\Omega_{0}\times\\{0\\}$, with $C^{1,\alpha}$ norm
tending to zero.

Theorem 1.1 is stated for codimension-one varifolds in Euclidean space, but
versions hold for higher-codimension varifolds and other ambient manifolds;
see Section 8.

We note that there are two classes of free-boundary planes: if
$\Omega=W^{2}\times\mathbb{R}^{n-1}$ is a wedge, then one has “horizontal”
planes containing $W^{2}\times\\{0\\}$, and “vertical” planes containing
$\\{0\\}\times\mathbb{R}^{n-1}$. Our Theorem holds only for horizontal planes,
and in general an Allard-type regularity fails for vertical planes (Example
3.6). Likewise, regularity as in Theorem 3.1 can fail when $\Omega$ is non-
convex (Example 3.7).

We also remark that, by considering the corresponding linear Neumann problem,
we expect $C^{1,\alpha}$ regularity to be sharp for general $\Omega$. For
example, when $\Omega$ is a wedge with opening angle $\theta_{0}$, then we
would expect no better regularity than $C^{1,\alpha}$ for
$\alpha=\frac{\pi}{\theta_{0}}-1$. If the opening angle is $\leq\pi/2$, then
one might expect $C^{2,\alpha}$ regularity. In the special case when
$\Omega=W^{2}\times\mathbb{R}$ is a $3$-dimensional wedge, then Theorem 1.1
was proven for minimal graphs over the free-boundary plane
$W^{2}\times\\{0\\}$ by [22].

Our regularity theorem implies a partial regularity for codimension-one
minimizing currents in domains which are locally modeled on polyedral cone
domains. The interior dimension bound of course follows from classical
interior regularity – our contributions are the estimates on the boundary. See
Section 9 for exact definitions and the proof.

######  Theorem 1.2 (Partial regularity).

Let $\Omega$ be a locally polyhedral $C^{2}$ domain in a complete $C^{3}$
Riemannian manifold $N^{n+1}$. Suppose $T=T\llcorner\mathrm{int}\Omega$ is a
free-boundary area-minimizing integral current in $\Omega$, or that
$T=\partial[U]\llcorner\mathrm{int}\Omega$ is a solution to the relative
isoperimetric problem, i.e. $T$ is area-minimizing in the class of
$T^{\prime}=\partial[U^{\prime}]\llcorner\mathrm{int}\Omega$ where
$U^{\prime}$ is relatively open in $\Omega$ and $|U^{\prime}|=|U|$.

Then $\dim(\mathrm{sing}T\cap\mathrm{int}\Omega)\leq n-7$, and:

  1. (1)

$\dim(\mathrm{sing}T\cap\partial\Omega)\leq n-2$, for general $\Omega$;

  2. (2)

$\dim(\mathrm{sing}T\cap\partial\Omega)\leq n-3$, if the dihedral angles of
$\Omega$ are $\leq\pi/2$ (in which case $\Omega$ is actually a domain-with-
corners);

  3. (3)

$\dim(\mathrm{sing}T\cap\partial\Omega)\leq n-7$, if the dihedral angles of
$\Omega$ are $=\pi/2$.

We emphasize that $\Omega$ need only be locally a $C^{2}$ perturbation of a
convex polyhedral domain, and does not need not be (locally) convex itself.

We think it should be true that $\dim(\mathrm{sing}T\cap\partial\Omega)\leq
n-7$ when the dihedral angles are $\leq\pi/2$. The stumbling block is proving
an appropriate Neumann eigenvalue bound for domains contained in an octant of
$S^{2}$. In this spirit, when the dihedral angles of $\Omega$ are $\leq\pi/2$
our proof in fact gives $\dim(\mathrm{sing}M\setminus\partial_{n-2}\Omega)\leq
n-7$, where $\partial_{n-2}\Omega$ consists of points where $\Omega$ is
locally modelled on domains with at least $n-2$ dimensions of symmetry. On the
other hand, it is plausible that the $n-2$ bound for general $\Omega$ is
sharp, as there are free-boundary planes in $3$-dimensional polyhedral cone
domains which we think may be minimizing (Example 9.5).

Our strategy is to prove a certain “excess decay” inequality (3.5). The two
key difficulties we encounter are a lack of reflection principle, and a lack
of a single boundary model. This means that, unlike regularity with smooth
boundaries (such as Allard [2], Gruter [13]), we do not have any easy
characterization of low-density tangent cones, nor do we have a nice monotone
quantity defined at all scales and all points along the boundary. Our proof
therefore relies comparatively little on monotonicity, in contrast to [2],
[13] who used it as a key tool to obtain among other things good effective
graphical approximations.

Instead, we prove a trace-like inequality for the first variation, and use
Moser iteration to prove good lower ahlfors regularity and non-concentration
estimates. The key to make our excess decay argument work is a sharp
$L^{\infty}$-$L^{2}$ bound, which implies that, even at the scale of the
excess, the $L^{2}$ norm cannot “accumulate” near the boundary. The main
technical challenge is establishing a first-variation control, and the
corresponding trace-like inequality. To do this we induct on the dimension of
the cone, basing our argument on work of Simon [34] who considered
$2$-dimensional capillary surfaces in a $3$-dimensional wedge.

While $\Omega$ resembles a fixed cone-type, we can prove excess decay by
comparison with the linear problem, in the spirit of DeGiorgi’s original proof
of interior regularity. However, inevitably at certain finite scales $\Omega$
will cease to look like a given cone, and here this argument breaks down (e.g.
in a wedge, every big ball looks like the wedge, but as the radius shrinks you
may look like a half-space, or an interior point). By an inductive argument on
the strata, we show that $\Omega$ looks like one of finitely-many cone types,
away from finitely many scales. By giving up some constant, we can chain
together our excess decays at each cone model to obtain a global excess decay
down to $0$. This last step is very general, and would apply to any regularity
problem in a “stratified” model.

It would be interesting to see if any of our techniques carry over to
regularity problems with other boundary conditions, for example the capillary
problem. De Philippis-Maggi [8] proved a (partial-)regularity theorem for
energy-minimizing capillary surfaces in $C^{1,1}$ domains, and the second
author [28] proved regularity of energy-minimizing $2$-dimensional capillary
surfaces in locally polyhedral Lipschitz domains. However very little is known
for capillary surfaces which are only stationary for area, nor is the sharp
regularity known for energy-minimizing capillary surfaces in higher
dimensions. We mention here that recently Kagaya-Tonegawa [24] proved a
monotonicity formula for stationary capillary surfaces.

Acknowledgements:  The authors thank Rick Schoen, Luca Spolaor, Brian White,
and Neshan Wickramasekera for helpful conversations. The authors also thank
the anonymous referee for suggestions on improvements of the paper. C.L. wants
to thank the Institute for Advanced Study, where part of this work was carried
out. N.E. was supported in part by NSF grant DMS-1606492. C.L. was supported
in part by NSF grant DMS-2005287.

##  2\. Notation and preliminaries

###  2.1. Notation

We work in $\mathbb{R}^{n+1}$. Given a subset $A\subset\mathbb{R}^{n+1}$, we
define $d_{A}(x)=\inf_{a\in A}|x-a|$ to be the usual Euclidean distance to
$A$. If $A=\emptyset$, we define $d_{A}(x)=\infty$. Write
$B_{r}(A)=\\{x:d_{A}(x)<r\\}$ for the open $r$-tubular neighborhood of $A$,
and write $\overline{A}$ for the closure of $A$. $B_{r}(x)$ is the open
$r$-ball centered at $x$. We write $B_{r}^{p}(x)\equiv
B_{r}(x)\cap\mathbb{R}^{p}\times\\{0^{n+1-p}\\}$, and $S^{p-1}\equiv\partial
B_{1}^{p}$ for the unit sphere in $\mathbb{R}^{p}\times\\{0^{n+p-1}\\}$.
Define the dilation/translation operator $\eta_{y,r}(x)=(x-y)/r$. Given a
subset $\Omega\subset\mathbb{R}^{n+1}$, write
$\Omega_{y,r}=\eta_{y,r}(\Omega)$.

Given vectors $v,w\in\mathbb{R}^{n+1}$, then $v\cdot w$ denotes the usual
Euclidean inner product, and $|v|=\sqrt{v\cdot v}$ is the usual Euclidean
length. Given linear maps $A,B:\mathbb{R}^{n+1}\to\mathbb{R}^{n+1}$, we define
the inner product $A\cdot B=\sum_{i}A(e_{i})\cdot B(e_{i})$, summed over any
choice of orthonormal basis $e_{i}$ of $\mathbb{R}^{n+1}$, and correspondingly
set $|A|^{2}=A\cdot A$. Unless otherwise stated, $e_{i}$ will denote the
standard basis of $\mathbb{R}^{n+1}$.

Given a subspace $V\subset\mathbb{R}^{n+1}$, we let $\pi_{V}$,
$\pi^{\perp}_{V}$ be the orthogonal projections to $V$, $V^{\perp}$
respectively. Given subpsaces $V$, $W$ (not necessarily of the same
dimension), then we set $V\cdot W=\pi_{V}\cdot\pi_{W}$. We remark that if $V$
and $W$ do have the same dimension, then

$\frac{1}{2}|\pi_{V}-\pi_{W}|^{2}=V^{\perp}\cdot W=V\cdot W^{\perp}.$

If $U\subset p+V$ and $f:U\to V^{\perp}$, we write
$\mathrm{graph}_{p+V}(f)=\\{x+f(x):x\in U\subset p+V\\}$. For ease of
notation, if $p+V=\mathbb{R}^{n}\times\\{0\\}$, and
$g:U\subset\mathbb{R}^{n}\times\\{0\\}\to\mathbb{R}$, then we interpret
$\mathrm{graph}_{\mathbb{R}^{n}\times\\{0\\}}(g)\equiv\mathrm{graph}_{\mathbb{R}^{n}\times\\{0\\}}(ge_{n+1})$.

Every constant written as $c$ or $c_{i}$ will be $\geq 1$, and every constant
$\delta_{i}$ or $\epsilon_{i}$ will be $\leq 1$. Unindexed constants may
change line-to-line.

###  2.2. Polyhedral cone domains

Here we define our model domains.

######  Definition 2.2.1.

A _polyhedral cone domain_ $\Omega$ in $\mathbb{R}^{n+1}$ is a closed,
dilation-invariant domain with non-empty interior, that can be written as the
intersection of finitely-many closed half-spaces.

Take a polyhedral cone domain $\Omega$ as defined above, and write
$\Omega=\cap_{j=1}^{N}H_{j}$ for $H_{j}=\\{x:x\cdot\nu_{j}\leq 0\\}$ being
closed half-spaces, and $\nu_{j}$ unit vectors. Here $\cap_{\emptyset}$ is
understood to be all of $\mathbb{R}^{n+1}$. If $\Omega$ contains a line $L$,
then $L\perp\nu_{j}$ for each $j$, and hence after a rotation we can write
$\Omega=\Omega^{\prime}\times\mathbb{R}$, where $\Omega^{\prime}$ is a
polyhedral cone domain in $\mathbb{R}^{n}$. We can therefore repeat this
process and find a maximal integer $m\in\\{0,\ldots,n+1\\}$ so that we can
write

$\Omega=O(\Omega_{0}^{l}\times\mathbb{R}^{m})$ (2.1)

for some $O\in SO(n+1)$, and $l+m=n+1$. Equivalently, we have
$O(\\{0\\}\times\mathbb{R}^{m})=\cap_{j}\partial H_{j}$ and
$O(\Omega_{0}\times\\{0\\})=(\cap_{j}\partial H_{j})^{\perp}$. Under this
decomposition $\Omega$ is said to be _$m$ -symmetric_. Note that $\Omega_{0}$
is $0$-symmetric, in the sense that it contains no lines.

We say $\Omega$ has dihedral angles $\leq\pi/2$ (resp. $=\pi/2$) if: given any
pair of half-spaces $H_{i},H_{j}$ such that $\partial H_{i}\cap\partial
H_{j}\cap\Omega$ is a relatively open subset of $\partial H_{i}\cap\partial
H_{j}$, then we have $\nu_{i}\cdot\nu_{j}\leq 0$ (resp. $=0$). We may refer to
$\Omega$ having dihedral angles $\leq\pi/2$ as _non-obtuse_.

######  Example 2.3.

If $L:\mathbb{R}^{n+1}\to\mathbb{R}^{n+1}$ is a linear isomorphism, then
$L([0,\infty)^{l}\times\mathbb{R}^{m})$ is a polyhedral cone domain. In fact
[7, Theorem 1.1] implies that any non-obtuse polyhedral cone domain is a
simplicial prism, and hence takes the form
$L([0,\infty)^{l}\times\mathbb{R}^{m})$ for some $l,m$ and $L$ as above.

######  Example 2.4.

When $l=0$, $\Omega=\mathbb{R}^{n+1}$. When $l=1$, $\Omega$ is a half-space.
When $l=2$, $\Omega$ is a wedge $W^{2}\times\mathbb{R}^{n-1}$, with interior
angle $<\pi$. The convex wedge $W^{2}\times\mathbb{R}^{n-1}$ is the archtype
of polyhedral cone domain, and $W^{2}\times\mathbb{R}^{n-2}$ is the archtype
of free-boundary minimal surface in this domain.

Given $x\in\Omega=\cap_{j=1}^{N}H_{j}$, there are indices
$I\subset\\{1,\ldots,N\\}$ so that we can write

$x\in\bigcap_{j\in I}\partial H_{j}\cap\bigcap_{j\not\in I}\mathrm{int}H_{j}.$
(2.2)

If we let $r<d(x,\cup_{i\not\in I}\partial
H_{i})\equiv\inf\\{|x\cdot\nu_{i}|:i\not\in I\\}$, then

$\Omega\cap B_{r}(x)=\cap_{i\in I}H_{i}\cap B_{r}(x)=\left(x+\cap_{i\in
I}H_{i}\right)\cap B_{r}(x).$ (2.3)

We therefore define the polyhedral cone domain $T_{x}\Omega:=\cap_{i\in
I}H_{i}$ to be the _tangent domain_ of $\Omega$ at $x$. Alternatively, one
could realize $T_{x}\Omega$ as the limit (in the local Hausdorff sense) of
dilations $\lim_{r\to 0}\frac{1}{r}(\Omega-x)$. Trivially
$T_{0}\Omega=\Omega$.

We define the _$i$ -th stratum_ $\partial_{i}\Omega$ to be the set of points
$x\in\Omega$, such that the tangent domain $T_{x}\Omega$ is _at most_
$i$-symmetric. Equivalently, if $x$ and $I$ are as in (2.2), then
$x\in\partial_{i}\Omega\iff\cap_{j\in I}\partial H_{j}$ is at most
$i$-dimensional. We have the trivial fibration

$\partial_{0}\Omega\subset\ldots\subset\partial_{n}\Omega=\partial\Omega\subset\partial_{n+1}\Omega=\Omega.$

For notational convenience define $\partial_{-1}\Omega=\emptyset$. If $\Omega$
is $m$-symmetric, in which case $\Omega=O(\Omega_{0}\times\mathbb{R}^{m})$ for
some $0$-symmetric $\Omega_{0}$, then we have
$\partial_{m}\Omega=O(\\{0\\}\times\mathbb{R}^{m})$,
$\partial_{i}\Omega=\emptyset$ for all $i<m$, and $T_{x}\Omega=\Omega$ for all
$x\in\partial_{m}\Omega$.

We shall use the following quantification of (2.3):

######  Lemma 2.5.

Given $\Omega$ a polyhedral cone domain, there is a number
$B(\Omega)\in(0,1/4)$ so that given $x\in\partial_{i}\Omega$, we have

$B_{2Bd}(x)\cap\Omega=B_{2Bd}(x)\cap(x+T_{x}\Omega),\quad
d=d(x,\partial_{i-1}\Omega).$

###### Proof.

Write $\Omega=\cap_{j=1}^{N}H_{j}$ as before, with
$H_{j}=\\{x:x\cdot\nu_{j}\leq 0\\}$. Fix an $i$. Let $\mathcal{I}_{i}$ be the
set of subcollections $I\subset\\{1,\ldots,N\\}$ with the property that
$V_{I}:=\cap_{j\in I}\partial H_{j}$ is an $i$-dimensional plane, and such
that $I$ is “maximal” in the sense that $\partial H_{k}\cap V_{I}$ is
$(i-1)$-dimensional for every $k\not\in I$. Take $I\in\mathcal{I}_{i}$. Let

$\alpha_{I}=\min\\{|\pi_{V_{I}}(\nu_{j})|:j\not\in I\\}.$

We have $\alpha_{I}>0$, as otherwise $\nu_{k}\perp V_{I}$ for some $k\not\in
I$, contradicting maximality of $I$.

Given $x\in V_{I}$ such that $d(x,V_{I}\cap\partial H_{k})\geq 1$ for every
$k\not\in I$, we have

$\displaystyle\min_{k\not\in I}d(x,\partial H_{k})$
$\displaystyle=\min_{k\not\in I}|x\cdot\nu_{k}|$
$\displaystyle\geq\alpha_{I}\min_{k\not\in
I}\frac{|x\cdot\pi_{V_{I}}(\nu_{k})|}{|\pi_{V_{I}}(\nu_{k})|}$
$\displaystyle=\alpha_{I}\min_{k}d(x,V_{i}\cap\partial H_{k})$
$\displaystyle\geq\alpha_{I}.$

Let $\alpha_{i}=\min_{I\in\mathcal{I}_{i}}\alpha_{I}$, which is positive since
$\mathcal{I}_{i}$ is finite.

When $i=0$ there is nothing to show, likewise if
$d(x,\partial_{i-1}\Omega)=0$. Take
$x\in\partial_{i}\Omega\setminus\partial_{i-1}\Omega$, and after scaling we
can WLOG suppose that $d(x,\partial_{i-1}\Omega)\geq 1$. Write

$x=\bigcap_{x\in I}\partial H_{j}\cap\bigcap_{x\not\in I}\mathrm{int}H_{j}$
(2.4)

as in (2.2). By our choice of $x$ we have that $\cap_{j\in I}\partial H_{j}$
is $i$-dimensional. If $k\not\in I$, then necessarily $\partial
H_{k}\cap(\cap_{j\in I}\partial H_{j})$ is $(i-1)$-dimensional, as otherwise
we would have $\partial H_{k}\cap(\cap_{j\in I}\partial H_{j})=\cap_{j\in
I}\partial H_{j}\ni x$, contradicting our decomposition (2.4). This implies
that if $y\in\partial H_{k}\cap(\cap_{j\in I}\partial H_{j})$, then
$y\in\partial_{i-1}\Omega$, and hence

$\min_{k\not\in I}d(x,V_{I}\cap\partial H_{k})\geq
d(x,\partial_{i-1}\Omega)\geq 1.$

Therefore $I\in\mathcal{I}_{i}$, and $x\in V_{I}$ satisfies
$d(x,V_{I}\cap\partial H_{k})\geq 1$ for every $k\not\in I$. We deduce by our
earlier computations

$\min_{k\not\in I}d(x,\partial H_{k})\geq\alpha_{i},$

and hence

$B_{\alpha_{i}}(x)\cap\Omega=B_{\alpha_{i}}(x)\cap(\cap_{j\in
I}H_{j})=B_{\alpha_{i}}(x)\cap(x+T_{x}\Omega)$

as in (2.3). Taking $B=\frac{1}{2}\min_{i=1,\ldots,n+1}\alpha_{i}$ proves the
Lemma. ∎

Let us define the density of $\Omega$ as

$\Theta_{\Omega}=\omega_{n+1}^{-1}\mathcal{H}^{n+1}(\Omega\cap
B_{1})\equiv\omega_{l}^{-1}\mathcal{H}^{l}(\Omega_{0}\cap B_{1}),$ (2.5)

where $\omega_{n}=\mathcal{H}^{n}(B_{1}(0^{n}))$ is the $n$-dimensional volume
of the unit $n$-ball. By convexity, we have the monotonicity

$\Theta_{T_{x}\Omega}\geq\frac{\mathcal{H}^{n+1}(\Omega\cap
B_{r}(x))}{\omega_{n+1}r^{n+1}}\quad\forall r>0,$ (2.6)

and hence we have the following lower-semi-continuity: if $x_{i}\to x$, then

$\liminf_{i}\Theta_{T_{x_{i}}\Omega}\geq\Theta_{T_{x}\Omega}.$ (2.7)

Let $\mathcal{N}_{\Omega}$ be the cone consisting of _outer normals_ for
$\Omega$:

$\mathcal{N}_{\Omega}=\cup\\{\nu\in\mathbb{R}^{n+1}:\Omega\subset\\{y:y\cdot\nu\leq
0\\}\\}.$

By convexity, $\mathcal{N}_{\Omega}\neq\emptyset$.

If we decompose $\Omega=O(\Omega_{0}^{l}\times\mathbb{R}^{m})$ as in (2.1),
then define $\mathcal{P}_{\Omega}$ to be the collection of _horizontal
$n$-planes_ of the form $O(\mathbb{R}^{l}\times W^{m-1})$ for $W^{m-1}$ an
$(m-1)$-plane in $\mathbb{R}^{m}$. We observe the trivial inclusion

$\mathcal{P}_{\Omega}\subset\mathcal{P}_{T_{x}\Omega}\quad\forall x\in\Omega.$

In particular, if $x\in\partial_{i}\Omega$ and $y\in\Omega\cap
B_{2Bd(x,\partial_{i-1}\Omega)}(x)$, then

$\mathcal{P}_{T_{x}\Omega}\subset\mathcal{P}_{T_{y}\Omega}.$ (2.8)

###  2.6. Curved polyhedral cones

Since a general polyhedral domain will at finite scales only look like the
perturbation of a polyhedral cone domain, in our local regularity we must
allow for domains with a little bit of curvature (captured by the map $\Phi$).
Moreover, since the model polyhedral cone will in general change as one moves
along any given stratum, we allow for small changes in the model domain itself
(captured by $\Psi$).

######  Definition 2.6.1.

Let ${\Omega^{(0)}}$ be a polyhedral cone domain in $\mathbb{R}^{n+1}$, as per
Definition 2.2.1. Given $\epsilon\in(0,1)$, we define
$\mathcal{D}_{\epsilon}({\Omega^{(0)}})$ as the set of domains $\Omega$
satisfying

$\Omega\cap B_{1}=\Phi(\Psi({\Omega^{(0)}}))\cap B_{1},$

where $\Psi$ is a linear isomorphism satisfying
$|\Psi-\mathrm{Id}|\leq\epsilon$, and $\Phi:B_{2}\to\mathbb{R}^{n+1}$ is a
$C^{2}$ diffeomorphism satisfying

$\Phi(0)=0,\quad
D\Phi|_{0}=\mathrm{Id},\quad|\Phi-\mathrm{id}|_{C^{2}(B_{2})}\leq\epsilon.$
(2.9)

######  Remark 2.7.

Since $\epsilon<1$, we have $\Phi(B_{2})\supset B_{1}$. In fact we could
equivalently have asked for $\Phi$ to be a diffeomorphism $B_{1}\to B_{1}$,
but we shall see our definition is slightly more convenient to work with.

Take $\Omega=\Phi(\Psi({\Omega^{(0)}}))\cap B_{1}$ as in Definition 2.6.1, and
given $x\in\Omega$ let us write $x=\Phi(\Psi(z))$. We define
$\partial_{i}\Omega=\Phi(\Psi(\partial_{i}{\Omega^{(0)}}))\cap B_{1}$, so that
$\partial_{i}\Omega$ consists of the points near which $\Omega$ is
diffeomorphic to some polyhedral cone that is at most $i$-symmetric.

There is a well-defined polyhedral cone domain $T_{x}\Omega=\lim_{r\to
0}\frac{1}{r}(\Omega-x)$ which we will call the tangent domain, and in fact we
can write $T_{x}\Omega=(D\Phi|_{\Psi(z)}\circ\Psi)T_{z}{\Omega^{(0)}}$. It
follows by scaling that if $\Omega\in\mathcal{D}_{\epsilon}({\Omega^{(0)}})$,
then $\frac{1}{R}\Omega\cap B_{1}\in\mathcal{D}_{\epsilon}({\Omega^{(0)}})$
for every $R\geq 1$, and hence
$T_{0}\Omega\in\mathcal{D}_{\epsilon}({\Omega^{(0)}})$ also.

Since $T_{x}\Omega$ is a polyhedral cone domain, we can define density
$\Theta_{T_{x}\Omega}$ and cone of outer normals $\mathcal{N}_{T_{x}\Omega}$
as before. We say a vector field $X$ is _tangential to $\Omega$_ if $X(x)\cdot
V=0$ for all $V\in\mathcal{N}_{T_{x}\Omega}$, and for all $x\in\Omega$.
Similar to (2.7), $\Theta$ obeys the following lower-semi-continuity:

######  Lemma 2.8.

Suppose $x_{i}\to x\in B_{1}$, and
$\Omega_{i}\in\mathcal{D}_{\delta_{i}}({\Omega^{(0)}})$ for some
$\delta_{i}\to 0$. Then we have

$\liminf_{i}\Theta_{T_{x_{i}}\Omega_{i}}\geq\Theta_{T_{x}{\Omega^{(0)}}}.$
(2.10)

###### Proof.

Write $\Omega_{i}=\Phi_{i}(\Psi_{i}({\Omega^{(0)}}))\cap B_{1}$ as per
Definition 2.6.1, and let $z_{i}=\Psi_{i}^{-1}(\Phi_{i}^{-1}(x_{i}))$. We have
$T_{x_{i}}\Omega_{i}=(D\Phi_{i}|_{\Psi_{i}(z_{i})}\circ\Psi_{i})(T_{z_{i}}{\Omega^{(0)}})$,
and therefore

$\displaystyle\Theta_{T_{x_{i}}\Omega}$
$\displaystyle\geq\omega_{n+1}^{-1}(1-c(n)\delta_{i})\mathcal{H}^{n+1}(T_{z_{i}}{\Omega^{(0)}}\cap
B_{1-c(n)\delta_{i}})$
$\displaystyle\geq(1-c(n)\delta_{i})\Theta_{T_{z_{i}}{\Omega^{(0)}}}.$

(2.10) then follows from the lower-semi-continuity (2.7). ∎

###  2.9. Varifolds

Our notion of weak surface will be a varifold. For a more detailed background
see [35] or [1]. Recall that an integral $n$-varifold in an open set $U$ is a
Radon measure $M$ on $U\times Gr(n,n+1)$ of the form

$M(\phi(x,S))=\int_{\tilde{M}}\phi(x,T_{x}M)\theta(x)d\mathcal{H}^{n}$

for some countably $n$-rectifiable set $\tilde{M}$, and some non-negative
Borel measurable function $\theta:\tilde{M}\to\mathbb{Z}$. Here $Gr(n,n+1)$
denotes the set of unoriented $n$-planes in $\mathbb{R}^{n+1}$. We write
$\mathcal{IV}_{n}(U)$ for the space of integral $n$-varifolds in $U$. If $S$
is an $n$-dimensional, $C^{1}$ submanifold of $\mathbb{R}^{n+1}$, we write
$[S]$ for the obvious varifold induced by $S$.

Given $M\in\mathcal{IV}_{n}(U)$, the mass measure $\mu_{M}=\pi_{\sharp}M$ is
the pushforward under the projection $\pi:U\times Gr\to U$, so that
$\mu_{M}=\mathcal{H}^{n}\llcorner\theta\llcorner\tilde{M}$. Given a
$C^{1}_{c}(U)$ vector field $X$, generating a $1$-parameter family of
diffeomorphisms $\phi_{t}:U\to\mathbb{R}^{n+1}$, the first-variation of $M$
along $X$ is the derivative

$\delta
M(X):=\frac{d}{dt}|_{t=0}(\phi_{t})_{\sharp}M=\int\mathrm{div}_{M}(X)d\mu_{M},$

where $div_{M}(X)=\sum_{i}e_{i}\cdot D_{e_{i}}X$, for any choice of ON basis
$\\{e_{i}\\}_{i}$ of $T_{x}\tilde{M}$. Relatedly, given a function $h\in
C^{1}_{c}(U)$, we write $\nabla h=\pi_{T_{x}\tilde{M}}(Dh)$ for the tangential
derivative of a function $h$ along $M$.

$M$ is said to have locally finite first variation if $\delta M$ is a bounded
operator on every $W\subset\subset U$. In this case we may decompose

$\delta M(X)=-\int X\cdot Hd\mu_{M}+\int X\cdot\eta d\sigma$

where $H$ is the generalized mean curvature of $M$, $\sigma$ the generalized
boundary measure, and $\eta$ the generalized boundary conormal.

We define the density ratio of $M$ in a ball $B_{r}(x)\subset U$ as

$\theta_{M}(x,r):=\frac{\mu_{M}(B_{r}(x))}{\omega_{n}r^{n}}.$

If $M$ has locally finite first variation, has zero generalized boundary, and
$||H_{M}||_{L^{\infty}(U;\mu_{M})}\leq\Lambda$, then $e^{\Lambda
r}\theta_{M}(x,r)$ is increasing in $0<r<d(x,\partial U)$ [1]. In particular,
the density at a point

$\theta_{M}(x):=\lim_{r\to 0}\theta_{M}(x,r)$

is a well-defined upper-semi-continuous function, satisfying
$\theta_{M}(x)\geq 1$.

Given a $C^{1}$ domain-with-corners $\Omega\subset\mathbb{R}^{n+1}$, we define
the set of _integral varifolds in $B_{1}$ with free-boundary in $\Omega$_,
denoted $\mathcal{IVT}_{n}(\Omega,B_{1})$, to be the set
$M\in\mathcal{IV}_{n}(B_{1})$ satisfying the conditions that
$M=M\llcorner\pi^{-1}(\mathrm{int}\Omega)$, and

$\delta M(X)=-\int X\cdot H^{tan}_{M}d\mu_{M}$

for all $X\in C^{1}_{c}(B_{1})$ tangential to $\Omega$, for some
$H_{M}^{tan}\in L^{1}_{loc}(B_{1},\mathbb{R}^{n+1};\mu_{M})$. In other words,
$M$ has mean curvature but no boundary “tangential” to $\partial\Omega$ (made
precise in Theorem 4.3).

###  2.10. Excess

Our mechanism to establish regularity is the decay of an appropriate excess
quantity. Given $\Omega=\Phi(\Psi({\Omega^{(0)}}))\cap B_{1}$ as in Definition
2.6.1, and an $n$-plane $V$, the full $L^{2}$ excess is:

$\displaystyle E_{\delta}(\Phi,M,p+V,x,r)$
$\displaystyle=\max\left\\{r^{-n-2}\int_{B_{r}(x)}d_{p+V}^{2}d\mu_{M},\right.$
$\displaystyle\quad\left.\delta^{-1}r^{2}||H_{M}||^{2}_{L^{\infty}(B_{1};\mu_{M})}+\delta^{-1}r^{2}|D^{2}\Phi|_{C^{0}(B_{2r}(x))}\right\\}.$

We will often abbreviate

$E(\Phi,M,p+V,x,r)=E_{1}(\Phi,M,p+V,x,r),$

and may also omit the $\Phi$ or $M$ when there is no ambiguity. At scales for
which $\Omega$ resembles a fixed model cone ${\Omega^{(0)}}$, we will prove
decay on $E_{\delta}$ by reducing the problem to a decay estimate of the
linearized problem.

When traversing cone types, we will find it convenient to work with the
“total” excess:

$\displaystyle E^{\infty}(M,p+V,x,r)$
$\displaystyle=r^{-2}\sup_{z\in\mathrm{spt}M\cap B_{r}(x)}d_{p+V}(z)^{2}$
$\displaystyle\equiv r^{-2}\sup_{z\in\mathrm{spt}M\cap
B_{r}(x)}|\pi_{V}^{\perp}(z-p)|^{2}$ $\displaystyle E^{W}(M,V,x,r)$
$\displaystyle=r^{-n}\int_{B_{r}(x)}|\pi_{T_{z}M}-\pi_{V}|^{2}d\mu_{M}(z)$
$\displaystyle E^{tot}(\Phi,M,p+V,x,r)$
$\displaystyle=\max\\{E^{\infty}(M,p+V,x,r),E^{W}(M,V,x,r),E(\Phi,M,p+V,x,r)\\}.$

Implicit in the definition of excess is the requirement that $\Phi(0)=0$,
$D\Phi|_{0}=\mathrm{Id}$, so even though $E$ is formally scale-invariant one
must be a little careful: in a general ball $B_{r}(x)\cap\Omega$ will not look
like a cone, and even when it does, if $x\neq 0$ then $\Phi$ will no longer be
the right map. This is made precise in the following section.

###  2.11. Changing cone type

The key fact that we will use in our regularity theorem is that at any point,
and at any appropriately small scale, $\Omega$ looks like one of finitely-many
polyhedral cone domains.

######  Lemma 2.12.

Let ${\Omega^{(0)}}$ be a polyhedral cone in $\mathbb{R}^{n+1}$. There is a
finite set of polyhedral cone domains
$\mathcal{T}({\Omega^{(0)}}):=\\{T_{z}{\Omega^{(0)}}:z\in{\Omega^{(0)}}\\}$,
and constants $B({\Omega^{(0)}})\in(0,1/4)$, $\epsilon_{B}(n)$, $c_{B}(n)$, so
that the following holds.

Given any $\Omega=\Phi(\Psi({\Omega^{(0)}}))\cap
B_{1}\in\mathcal{D}_{\epsilon}({\Omega^{(0)}})$, for
$\epsilon\leq\epsilon_{B}$, and given $x=\Phi(\Psi(z))\in\partial_{i}\Omega$,
take $r\leq\min\\{Bd(x,\partial_{i-1}\Omega),1-|x|\\}$, and define
$\Omega_{x,r}=\frac{1}{r}(\Omega-x)$. Then the following holds:

  1. (1)

We have

$(1-c_{B}\epsilon)d(x,\partial_{i-1}\Omega)\leq
d(z,\partial_{i-1}{\Omega^{(0)}})\leq(1+c_{B}\epsilon)d(x,\partial_{i-1}\Omega),$
(2.11)

  2. (2)

Given $y\in\Omega\cap B_{r}(x)$, then for every
$V\in\mathcal{P}_{T_{x}\Omega}$ there is a $W\in\mathcal{P}_{T_{y}\Omega}$
such that $|\pi_{V}-\pi_{W}|\leq c(n)|D^{2}\Phi|_{C^{0}(B_{2})}|x-y|$.

  3. (3)

There is a $T_{z}{\Omega^{(0)}}\in\mathcal{T}$, a linear isomorphism
$\beta:\mathbb{R}^{n+1}\to\mathbb{R}^{n+1}$, a $C^{2}$ diffeomorphism
$\alpha:B_{2}\to\mathbb{R}^{n+1}$, satisfying

$|\beta-\mathrm{Id}|\leq c_{B}\epsilon,$ (2.12)

and $\alpha(0)=0$, $D\alpha|_{0}=\mathrm{Id}$, and

$(1-c_{B}\epsilon)|D^{2}\alpha|_{C^{0}(B_{2})}\leq
r|D^{2}\Phi|_{C^{0}(B_{2r}(x))}\leq(1+c_{B}\epsilon)|D^{2}\alpha|_{C^{0}(B_{2})},$
(2.13)

so that

$\Omega_{x,r}\cap B_{1}=\alpha(\beta(T_{z}{\Omega^{(0)}}))\cap B_{1},$ (2.14)

In particular,
$\Omega_{x,r}\in\mathcal{D}_{c_{B}\epsilon}(T_{z}{\Omega^{(0)}})$.

  4. (4)

In the notation of part 3,

$\displaystyle\frac{1}{2}E(\Phi,M,p+V,x,r)$ $\displaystyle\leq
E(\alpha,(\eta_{x,r})_{\sharp}M,\eta_{x,r}(p)+V,0,1)$ (2.15)
$\displaystyle\leq 2E(\Phi,M,p+V,x,r),$

and the same with $E^{tot}$ in place of $E$.

######  Remark 2.13.

If $x=0$, then for any $r\leq 1$ we have $r^{-1}\Omega\cap
B_{1}=\Phi_{0,r}(\Psi({\Omega^{(0)}}))\cap B_{1}$, where
$\Phi_{0,r}(y)=r^{-1}\Phi(ry)$, and therefore we have the exact scaling

$E(\Phi,M,p+V,0,r)=E(\Phi_{0,r},(\eta_{0,r})_{\sharp}M,r^{-1}p+V,0,1).$

###### Proof.

For $\epsilon_{B}(n)$ sufficiently small, $\Phi\circ\Psi$ is a
$(1+c(n)\epsilon)$-bi-Lipschitz equivalence:

$|\Phi(\Psi(y))-\Phi(\Psi(y^{\prime}))-(y-y^{\prime})|\leq
c(n)\epsilon|y-y^{\prime}|,$

from which 1 follows directly. In particular note that
$\Psi^{-1}\Phi^{-1}B_{r}(x)\subset
B_{2Bd(z,\partial_{i-1}{\Omega^{(0)}})}(z)$.

We prove 2. Observe that for a fixed subspace $U^{n}$, and linear isomorphisms
$A,B:\mathbb{R}^{n+1}\to\mathbb{R}^{n+1}$, the map $A\mapsto\pi_{A(U)}$ is
well-defined and analytic in $A$, and satisfies

$|\pi_{A(U)}-\pi_{B(U)}|\leq c(n)|A-B|$ (2.16)

for $A,B$ satisfying $|A-Id|+|B-Id|<\epsilon(n)$. Write $y=\Phi(\Psi(w))$.
From 1 and the inclusion (2.8), we have

$\mathcal{P}_{T_{z}{\Omega^{(0)}}}\subset\mathcal{P}_{T_{w}{\Omega^{(0)}}},$

and therefore we can take $W=D\Phi|_{w}\circ D\Phi^{-1}|_{x}V$. We then
estimate, provided $\epsilon_{B}(n)$ is sufficiently small,

$|\pi_{V}-\pi_{W}|\leq c(n)\left|D\Phi|_{w}\circ D\Phi^{-1}|_{x}-Id\right|\leq
c(n)\left|D\Phi^{-1}|_{x}-D\Phi^{-1}|_{y}\right|\leq
c(n)|x-y||D^{2}\Phi|_{C^{0}(B_{1})}.$

This proves 2.

We prove 3, 4. Let
$\mathcal{T}=\\{T_{z}{\Omega^{(0)}}\\}_{z\in{\Omega^{(0)}}}$, and $B$ be as in
Lemma 2.5. Let us define

$\beta=D\Phi|_{\Psi(z)}\circ\Psi,\quad\alpha(y)=\frac{1}{Br}(\Phi(x+rD\Phi^{-1}|_{x}y)-x).$

The bound (2.12) and $\alpha(0)=0$, $D\alpha|_{0}=\mathrm{Id}$ follow
trivially. The Hessian bound in (2.13) follows from our definition of $\alpha$
and the estimate
$(1-c(n)\epsilon)|v|\leq|D\Phi|_{y}v|\leq(1+c(n)\epsilon)|v|$. Ensuring
$\epsilon_{B}(n)$ is small, we have

$r\leq 2Bd(z,\partial_{i-1}{\Omega^{(0)}}),$

and therefore

$\displaystyle(\alpha\circ\beta)(B_{2}\cap T_{x}{\Omega^{(0)}})\cap B_{1}$
$\displaystyle=(\alpha\circ\beta)(B_{2}\cap{\Omega^{(0)}}_{x,r})\cap B_{1}$
$\displaystyle=\Omega_{x,r}\cap B_{1}.$

This proves 3. Lastly, 4 follows by (2.13) and scaling, ensuring
$\epsilon_{B}(n)$ is sufficiently small. ∎

##  3\. Main theorem

Our main Theorem 3.1 is the following Allard-type regularity result, which
says loosely that whenever an integral varifold $M$ has free-boundary in
$\Omega=\Phi({\Omega^{(0)}}={\Omega^{(0)}_{0}}\times\mathbb{R})$, and: $\Phi$
is sufficiently close to the identity, $M$ has sufficiently small mean-
curvature, and $M$ is sufficiently varifold close to the “horizontal” plane
$\mathbb{R}^{n}\times\\{0\\}$, then $\mathrm{spt}M$ is a $C^{1,\alpha}$
perturbation of ${\Omega^{(0)}_{0}}$.

In general $\Omega$ is curved, and so $\mathrm{spt}M$ will not be graphical
over a particularly “nice” subdomain of $\mathbb{R}^{n}$. Instead, it is more
convenient and precise to look at $\Phi^{-1}(\mathrm{spt}M)$, which will be a
graph over $\mathbb{R}^{n}\cap{\Omega^{(0)}}\cap B_{1/32}$. We do not lose
anything in our estimates by doing this, as even before this transformation we
must use $|D^{2}\Phi|_{C^{0}}$ to control the tilting of tangent planes of
$\mathrm{spt}M$.

For various reasons we in fact want to allow not only the diffeomorphism
$\Phi$ to change, but also the reference domain ${\Omega^{(0)}}$ (mainly
because different points in $\Phi({\Omega^{(0)}})$ will be modeled on
different polyhedral cone domains, even when staying in the same stratum; see
Section 2.11). For this reason we in fact consider domains of the form
$\Phi(\Psi({\Omega^{(0)}}))$, where $\Phi$ is a diffeomorphism close to $Id$,
and $\Psi$ is a linear map close to $Id$. Our constants $\delta,c,\alpha$ will
be uniform in $\Phi,\Psi$, but in each particular case our “reference”
polyhedral domain will be $\Psi({\Omega^{(0)}})$. It may be easier to parse
Theorem 3.1 by considering the case when $\Psi=Id$, $p=0$, $V=\mathbb{R}^{n}$,
in which case $q=0$, $W=\mathbb{R}^{n}$, $T_{0}\Omega={\Omega^{(0)}}$.

######  Theorem 3.1 (Allard-type regularity).

Let ${\Omega^{(0)}}$ be a polyhedral cone domain. There are constants
$\delta({\Omega^{(0)}})$, $c({\Omega^{(0)}})$, $\alpha({\Omega^{(0)}})$ so
that the following holds. Let $\Omega=\Phi(\Psi({\Omega^{(0)}}))\cap
B_{1}\in\mathcal{D}_{\delta}({\Omega^{(0)}})$, and take
$M\in\mathcal{IVT}(\Omega,B_{1})$. Assume there is a
$V\in\mathcal{P}_{T_{0}\Omega}$, $p\in V^{\perp}$, so that

$E:=\int_{B_{1}}d_{p+V}^{2}d\mu_{M}+||H||_{L^{\infty}(B_{1};\mu_{M})}^{2}+|D^{2}\Phi|_{C^{0}(B_{2})}^{2}\leq\delta^{2}$
(3.1)

and

$\theta_{M}(0,1)\leq(3/2)\Theta_{T_{0}\Omega},\quad\mathrm{spt}M\cap
B_{1/512}\neq\emptyset.$ (3.2)

Then if we set $q=\Phi^{-1}(p)$, $W=D\Phi^{-1}|_{p}V$, we can find a function
$f:(q+W)\cap\Psi({\Omega^{(0)}})\cap B_{1/128}(q)\to W^{\perp}$ satisfying

$|f|_{C^{1,\alpha}}\leq cE^{1/2},$ (3.3)

so that

$\Phi^{-1}(\mathrm{spt}M)\cap B_{1/256}\subset\Phi^{-1}(\mathrm{spt}M)\cap
B_{1/128}(q)=\mathrm{graph}_{q+W}(f)\cap B_{1/128}(q).$ (3.4)

Some comments are in order.

######  Remark 3.2.

Even though ${\Omega^{(0)}}$ is convex, $\Omega$ need not be.

######  Remark 3.3.

$\alpha$ can in fact be chosen to be anything in some interval $(0,e-1)$
(where $e=e({\Omega^{(0)}})>1$ as in Proposition 5.3 is determined by the
Neumann eigenvalue expansion of $\Omega_{0}^{l}\cap\partial B_{1}$), provided
$\delta$ and $c$ are taken to depend on $\alpha$. For example, when
${\Omega^{(0)}}$ is $\mathbb{R}^{n+1}$ or a half-plane, then any
$\alpha\in(0,1)$ is admissible. When
${\Omega^{(0)}}=W^{2}\times\mathbb{R}^{n-1}$ is a wedge with angle
$\gamma<\pi$, then we can take $\alpha\in(0,\min\\{1,\pi/\gamma-1\\})$. See
Remark 5.4 for more details.

######  Remark 3.4.

We state and prove Theorem 3.1 in codimension-one Euclidean space. However
when $l=1$ our proof carries over verbatim to higher codimension and ambient
manifolds, giving an alternate proof of [14]. When $l\geq 2$, the proof
carries over except for two estimates in Section 4, which continue to hold if
one knows a priori that $\mathrm{spt}M$ is contained in some
$(n+1)$-dimensional submanifold. See Section 8 for details.

######  Remark 3.5.

If one assumes $\theta_{M}(0,1)\leq(1+\delta)\Theta_{T_{0}\Omega}$ then
Theorem 3.1 holds for varifolds which are only rectifiable, but have a lower
density bound $\theta_{M}\geq 1$ $\mu_{M}$-a.e. This requires only minor
modifications of the proof. (Specifically, in the contradiction arguments of
Proposition 6.4 and Lemma 7.2, the choice of constant $\delta_{1}$ in
Corollary 6.2, and the choice of constants in the induction argument of of
Theorem 7.1.). In a similar vein, Theorem 3.1 also holds if we assume
$H_{M}\in L^{p}(\mu_{M})$, for $p>n$ instead of $p=\infty$. In this case our
constants would depend on $p$ also.

Our regularity Theorem 3.1 requires $V$ to be “horizontal,” in the sense that
if $\Psi({\Omega^{(0)}})={\Omega^{(0)}_{0}}\times\mathbb{R}$, then
$V\supset{\Omega^{(0)}_{0}}$. When $V$ is instead “vertical,” in the sense
that $V\supset\\{0\\}\times\mathbb{R}$, then regularity as in Theorem 3.1 can
fail. Below is an example illustrating this.

######  Example 3.6.

Let $W\subset\mathbb{R}^{2}$ be the wedge
$\\{re^{i\theta}:r\in[0,\infty),-\pi/6\leq\theta\leq\pi/6\\}$, and let
$\mathbf{Y}\subset\mathbb{R}^{2}$ be the cone over
$\\{1,e^{i2\pi/3},e^{-i2\pi/3}\\}$, consisting of three rays meeting at the
origin at $120^{0}$. Then for every $\epsilon>0$, the integral varifold
$M_{\epsilon}$ given by integrating over $(\epsilon+\mathbf{Y})\cap W$ is
stationary with free-boundary in $W$. As $\epsilon\to 0$, then $M_{\epsilon}$
converges as varifolds to the “vertical” plane $P=\\{x\geq 0,y=0\\}$, but none
of the $M_{\epsilon}$ are $C^{1}$ perturbations of $P$.

One can construct a similar example by restricted the tetrahedal cone to a
$0$-symmetric domain in $\mathbb{R}^{3}$ consisting of the intersection of $4$
half-spces.

Example 3.6 is a little contrived, but we expect one should be able to
construct smooth counterexamples. However, we would not expect these examples
to be minimizing (in the sense of integral currents). Relatedly, when
${\Omega^{(0)}}$ is a wedge, or has dihedral angles $\leq\pi/2$, then the
vertical planes are not minimizing (Lemma 9.4). For more general convex
${\Omega^{(0)}}$, or when ${\Omega^{(0)}}$ is a non-convex wedge, this may
fail (see Examples 3.7, 9.5).

When $\Omega$ is non-convex, Theorem 3.1 can fail also, even when $M$ is in
some sense minimizing.

######  Example 3.7.

[21, Theorem 1] implies the following: Suppose $\Omega=W^{2}\times\mathbb{R}$,
where $W^{2}$ is a wedge with angle $>\pi$, and $\Gamma$ is any smooth curve
in $\mathrm{int}\Omega\cap\\{x^{2}+y^{2}=1\\}$, such that $\Gamma$ meets
$\partial\Omega$ only at its endpoints $\\{p_{1},p_{2}\\}$, and the height
function $x_{3}|_{\Gamma}$ has no maxima away from the endpoints. Let
$B_{+}=\\{(x,y)\in\mathbb{R}^{2}:x^{2}+y^{2}<1,y>0\\}$. Then there is a map
$F\in C^{0}(\overline{B_{+}},\Omega)\cap W^{1,2}(\overline{B_{+}},\Omega)$
such that

  1. (1)

$F$ minimizes the Dirichlet energy, and $F(B_{+})$ is a smooth minimal
surface;

  2. (2)

$F$ maps the semi-circle $\partial B_{+}\cap\\{y>0\\}$ monotonically to
$\Gamma$, $\pi_{\mathbb{R}^{2}\times\\{0\\}}\circ F$ maps the interval
$[-1,1]$ monotonically into to $\partial W$, and $\pi_{\mathbb{R}^{2}}\circ F$
maps $B_{+}$ diffeomorphically to $\mathrm{int}\Omega\cap B_{1}$;

  3. (3)

writing $F(\overline{B_{+}})\cap\\{0\\}\times\mathbb{R}=\\{q\\}$, then
$F^{-1}(q)=[a_{1},a_{2}]$ is an interval of positive length, and $F$ extends
smoothly to $\overline{B_{+}}\setminus\\{-1,1,a_{1},a_{2}\\}$;

  4. (4)

on $(-1,1)\setminus[a_{1},a_{2}]$, $F$ meets $\partial\Omega$ orthogonally;

  5. (5)

on $(a_{1},a_{2})$, the unit normal of $F$ is horizontal.

These items imply that
$M=[F(\overline{B_{+}})]\in\mathcal{IVT}_{2}(\Omega,\mathbb{R}^{3}\setminus\Gamma)$,
and has zero mean curvature. By choosing $\Gamma$ to be contained in a very
thin slab $\mathbb{R}^{2}\times[-\epsilon,\epsilon]$, the maximum principle
implies $F(\overline{B_{+}})$ is contained in this slab also. Therefore we can
arrange so that $M\llcorner B_{1/2}$ is arbitrarily varifold close to the
multiplicity-one horiztonal plane $[\mathbb{R}^{2}]\llcorner B_{1/2}$, but
$\mathrm{spt}M$ will never be graphical over $\mathbb{R}^{2}$ at $0$.

###  3.8. Outline of proof

Our strategy to prove Theorem 3.1 is to show the following excess decay
estimate: for all $x\in\mathrm{spt}M\cap B_{1/16}$ there is a plane
$V_{x}\in\mathcal{P}_{T_{x}\Omega}$ so that

$\sup_{z\in B_{r}(x)\cap\mathrm{spt}M}r^{-2}d(z,x+V_{x})^{2}\leq
c({\Omega^{(0)}})r^{2\alpha}E\quad\forall 0<r<1/4,$ (3.5)

where $\alpha({\Omega^{(0)}})\in(0,1)$. From (3.5) it follows easily that
$\mathrm{spt}M\cap B_{1/32}$, and hence $\Phi^{-1}(\mathrm{spt}M)\cap
B_{1/64}$, is a $C^{1,\alpha}$ graph with norm controlled by $cE^{1/2}$.

We prove (3.5) in two steps. In step one (Section 6), given any fixed model
cone ${\Omega^{(0)}}$, we prove a decay like (3.5) with $x=0$. Loosely
speaking we show that $\mathrm{spt}M\cap B_{1}$ resembles a $W^{1,2}$ harmonic
function in ${\Omega^{(0)}}$ with Neumann boundary conditions. By
understanding these linear solutions (by a Fourier expansion and an eigenvalue
estimate), we can prove $C^{1,\alpha}$ decay. This basic idea goes back to
DeGiorgi, who proved interior regularity by a similar “excess decay” strategy,
and is implemented in a fashion closer to our style in [1], [33], [10].

In step two (Section 7), we exploit the polyhedral structure of
${\Omega^{(0)}}$ to show that for every $x\in B_{1/4}$, and $0<r<1/4$, we can
find radii $r=r_{0}^{+}\geq r_{0}^{-}\geq r_{1}^{+}\geq r_{1}^{-}\ldots\geq
r_{n+1}^{+}\geq r_{n+1}^{-}=0$, such that when $s\in[r_{i}^{-},r_{i}^{+}]$
then $\Omega\cap B_{s}(x)$ is modelled on some $T_{z_{i}}{\Omega^{(0)}}$, and
$r_{i}^{-}/r_{i-1}^{+}\geq 1/c({\Omega^{(0)}})$. In other words, outside of
finitely many scales (controlled only by ${\Omega^{(0)}})$, $\Omega\cap
B_{s}(x)$ is modelled on some polyhedral cone of the form
$T_{z}{\Omega^{(0)}}$. Since there are only finitely many tangents
$T_{z}{\Omega^{(0)}}$, we can therefore inductively prove decay by our first
step in each interval $[r_{i}^{-},r_{i}^{+}]$, and extend decay from one
interval to the next by enlarging our constant $c$ in (3.5) by a controlled
amount.

The key technique hurdle in proving both steps is to show that $M$ has
controlled first variation $\delta M$. Our hypotheses imply $\delta M$ is only
bounded in directions tangential to $\Omega$, but we need to establish both
that $\delta M$ is bounded in all directions, and an a priori trace-like
estimate for $||\delta M||$ in terms of $||M||$ (Theorem4.3). This is the main
point where we use convexity of ${\Omega^{(0)}}$.

A priori control on $\delta M$ gives good compactness for sequences of such
$M$, and allows us to prove a sharp $L^{\infty}-L^{2}$ estimate, and a uniform
lower density bound. The $L^{\infty}$ estimate is crucial in Step 1, to show
that $M$ doesn’t “concentrate” near $\partial\Omega$ at the scale of excess
(and therefore can be well approximated by the interior “graphical” region).
The lower density bound means that in various (blow-up) limits the varifolds
do not disappear, as they might in the non-convex case.

We elaborate on these steps below. For simplicity, in our outline we will
assume $\Omega={\Omega^{(0)}}$, and $H=0$.

Step 1: Decay towards a single cone model. We wish to prove an excess decay of
the following type: if $M$ is sufficiently varifold close in $B_{1}$ to a
horizontal plane $(p+V)\cap\Omega$, then there is a new horizontal plane
$p^{\prime}+V^{\prime}$ and a $\theta(\Omega)$ so that

$E(M,p^{\prime}+V^{\prime},0,\theta)\leq\frac{1}{2}E(M,p+V,0,1).$ (3.6)

Here $E(M,p+V,x,r)=r^{-n-2}\int_{B_{r}(x)}d_{p+V}(z)^{2}d\mu_{M}(z)$.
(Proposition 6.1 is phrased in terms of the $L^{2}$ excess, which is a little
more convenient, but in this setting the $L^{\infty}$ and $L^{2}$ excesses are
the same; see 4.7). By iterating (3.6), we obtain an estimate like

$E(M,p^{\prime\prime}+V^{\prime\prime},0,r)\leq
c(\Omega)r^{2\alpha}E^{\infty}(M,p+V,0,1)\quad\forall d(\mathrm{spt}M,0)\leq
r\leq 1.$

This is our main decay estimate in Step 1.

We prove (3.6) by contradiction. We assume there is a sequence of $M_{i}$ such
that $M_{i}\to[(p+V)\cap\Omega]$ in $B_{1}$ as varifolds, such that

$\inf_{p^{\prime}+V^{\prime}}E(M_{i},p^{\prime}+V^{\prime},0,\theta)\geq\frac{1}{2}E(M_{i},p+V,0,1/2)=:E_{i}\quad(\to
0\text{ as }i\to\infty).$

On larger and larger sets $U_{i}\subset\subset\mathrm{int}\Omega\cap(p+V)$
(for $U_{i}\to\mathrm{int}\Omega\cap(p+V)$) we can write
$\mathrm{spt}M_{i}\cap U_{i}=\mathrm{graph}_{p+V}(u_{i})$, where $u_{i}$
satisfy the minimal surface equation. By setting $v_{i}=E_{i}^{-1/2}u_{i}$,
then the $v_{i}$ are locally bounded in $L^{\infty}$, and after passing to a
subsequence we get convergence

$v_{i}\to v,$

where the _Jacobi field_ $v$ satisfies

$\Delta
v=0,\quad\partial_{n}v=0,\quad\int_{B_{1}(0)\cap(p+V)\cap\Omega}v^{2}\leq 1.$

We now prove two key technical facts: First, by the sharp $L^{\infty}$ bound
(4.31), we have strong convergence

$E_{i}^{-1}E(M_{i},p+V,0,r)\to
r^{-n-2}\int_{B_{r}(0)\cap(p+V)\cap\Omega}v^{2}\quad\forall 0<r<1/4.$

Second, the $W^{1,2}$ estimate of (4.32), a sharp eigenvalue estimate for
convex domains in the sphere (Theorem 5.1), and a standard Fourier-type
decomposition, we can expand $v$ as above like

$v=a+b\cdot x+O(r^{1+\alpha}),$

where $b$ lies in some direction of translational symmetry of $\Omega$. In
other words, the eigenvalue bound of Theorem 5.1 implies that any free-
boundary plane is integrable through rotations.

We can now repeat the blow-up argument with
$p_{i}+V_{i}=\mathrm{graph}_{p+V}(E_{i}a+E_{i}b\cdot x)$ in place of $p+V$,
and obtain the Jacobi field

$v^{\prime}=v-a-b\cdot x=O(r^{1+\alpha}).$

Hence for $i>>1$ we have

$E_{i}^{-1}E(M_{i},p_{i}+V_{i},0,r)\leq
2r^{-n-2}\int_{B_{r}(0)\cap(p+V)\cap\Omega}v^{2}\leq c(\Omega)r^{2\alpha},$

which is a contradiction for $r(\Omega)$ small.

This general strategy is very robust, and has been implemented in many other
contexts. However in any given situation there are typically two key technical
issues to address: strong convergence in norm of the non-linear problem to the
linear problem; and decay of the linear problem. We handle these in our
situation by our sharp $L^{\infty}$ bound and our sharp eigenvalue estimate.

Step 2: Decay across cone models. We prove the general decay (3.5) by using
Step 1 and an inductive argument on the strata. We first observe (Lemma 2.12)
that every point in $\Omega$ is locally modelled on some (other) polyhedral
cone. Precisely, there is a $B(\Omega)$ so that if $x\in\partial_{i}\Omega$,
and $r\leq d(x,\partial_{i-1}\Omega)$, then

$B_{Br}(x)\cap\Omega=B_{Br}(x)\cap(x+T_{x}\Omega).$

(When $\Omega\in\mathcal{D}_{\delta}({\Omega^{(0)}})$ is only a perturbation
of a polyhedral cone a similar statement holds.)

Now given $x\in\partial_{i}\Omega$, we choose points
$\tilde{x}_{j}\in\partial_{i_{j}}\Omega$ and radii

$1=r^{-}_{J+1}\geq r^{+}_{J}\geq r^{-}_{J}\geq\ldots\geq r_{0}^{+}\geq
r_{0}^{-}=0,$

so that $\Omega\cap B_{r}(\tilde{x}_{j})$ is modelled on some fixed polyhedral
cone $T_{z_{j}}{\Omega^{(0)}}$, when $r\in[r_{j}^{-},r_{j}^{+}]$, and
$r_{j}^{-}/r_{j-1}^{+}\geq 1/c({\Omega^{(0)}})$. The degree of symmetry
$i_{j}$ is strictly decreasing as $j$ increases, so $J\leq n$. We will apply
Step 1 to get decay $r_{j}^{+}\to r_{j}^{-}$, and then give up a controlled
number of scales to “decay” $r_{j}^{-}\to r_{j-1}^{+}$.

Some care must be taken in construting these points/radii, since for each $j$
we need that:

  1. (1)

$r_{j}^{+}\leq Bd(\tilde{x}_{j},\partial_{i_{j}-1}\Omega)$, so that
$\Omega\cap B_{r_{j}^{+}}(\tilde{x}_{j})$ looks close to a polyhedral cone;

  2. (2)

$|x-\tilde{x}_{j}|\leq r_{j}^{-}$, so that we can apply Step 1 in
$B_{r_{j}^{+}}(x_{j})$ to get decay down to $r_{j}^{-}$;

  3. (3)

$B_{r_{j}^{+}}(\tilde{x}_{j})\subset B_{r_{j+1}^{-}/2}(\tilde{x}_{j+1})$, but
$r_{j}^{+}\geq cr_{j+1}^{-}$, so that we can control the mass and excess in
$B_{r_{j}^{+}}(\tilde{x}_{j})$ in terms of the mass/excess in
$B_{r_{j+1}^{-}}(\tilde{x}_{j+1})$.

Once we construct these $\tilde{x}_{j}$, $r_{j}^{\pm}$, we inductively prove a
statement of the form: for each $j$, there is a
$V_{j}\in\mathcal{P}_{T_{\tilde{x}_{j}}\Omega}$ so that

$E(M,p_{j}+V_{j},\tilde{x}_{j},r)\leq\Lambda_{j}Er^{2\alpha}\quad\forall
r_{j}^{-}\leq r\leq 1/4,$

and

$\theta_{M}(\tilde{x}_{j},r)\leq(7/4)\Theta_{T_{\tilde{x}_{j}}\Omega}\quad\forall
r_{j}^{-}\leq r\leq r_{j}^{+}.$

The excess control ensures $M\llcorner B_{r}(\tilde{x}_{j})$ looks like a
plane with some multiplicity, the mass control ensures this multiplicity is
$\leq 1$, and the fact that $x\in B_{r_{j}^{-}/2}(\tilde{x}_{j})$ ensures this
multiplicity is $\geq 1$. The proof of this statement follows in a fairly
straightforward way from Lemma 2.12 and Step 1. An extra argument by
contradiction (Lemma 7.2) is required to ensure the mass control at scale
$B_{r_{j}^{-}}(\tilde{x}_{j})$ carries over to scale
$B_{r_{j-1}^{+}}(\tilde{x}_{j-1})$.

Minimizers Here we apply our regularity theorem to minimizing
currents/isoperimetric sets of finite perimeter. We classify low-dimensional
minimizing cones, and prove a compactness theorem, which together with our
regularity result implies a partial regularity theorem by standard dimension
reducing techniques.

The compactness is fairly straightforward – we just need to adapt an argument
of Gruter [13] to ensure mass cannot accumulate near the boundary. The main
step is classifying low-dimensional minimizing cones, which we prove by
induction on the number of symmetries of $\Omega$. The idea is as follows:
Assume that any minimizing $T$ in $\Omega_{0}\times\mathbb{R}^{m-1}$ is a
horizontal plane. Then if $T$ is minimizing in
$\Omega_{0}\times\mathbb{R}^{m}$, by induction and our regularity theorem $T$
is a $C^{1,\alpha}$ surface away from $0$. Under certain circumstances, then
we can boost this to $C^{2,\alpha}$ regularity, and thereby adapt Simons’
classical argument to prove $T$ must be planar. Then, again in certain
circumstances, a cut-and-paste argument implies this plane must be horizontal.

Unfortunately, even in low dimensions we start running into issues. The
barrier to adapting Simons’ argument when $\Omega_{0}$ is more than $3$
dimensional is the lack of $C^{2,\alpha}$ regularity in $3$-dimensional cones.
We suspect this holds if the dihedral angles are at most $\pi/2$, but this
requires a Neumann eigenvalue estimate for spherical domains which is not
known. On the flip side, for general dihedral angles it is not clear that
every minimizing plane need be horizontal – when $\Omega_{0}$ is only an
intersection of $3$ half-spaces in $\mathbb{R}^{3}$, it is plausible there are
non-horizontal minimizers. Taken together, we get a codimension $2$ bound for
the singular set in general domains, a codimension $3$ bound in domains with
dihedral angles $\leq\pi/2$, and the (sharp) codimension $7$ bound in certain
special cases (e.g when all dihedral angles are $=\pi/2$).

##  4\. First variation, mass control

We prove in this section that the mass and total first variation of an
$M\in\mathcal{IVT}_{n}(\Omega,B_{1})$ are controlled by the mass, tangential
mean curvature of $M$, and geometry of $\Omega$. We must first prove
monotonicity and the mass control, by cooking up an appropriate tangential
vector field, and then we use this to prove control on the first variation.
Two important consequences are lower Ahlfors-regularity of $M$ (Corollary
4.6), and $L^{\infty}-L^{2}$, $W^{1,2}-L^{2}$ bounds on excess (Corollary
4.7).

In this section we fix ${\Omega^{(0)}}$ a polyhedral cone domain (as per
Definition 2.2.1), and after a rotation there is no loss in assuming
${\Omega^{(0)}}={\Omega^{(0)}_{0}}^{l}\times\mathbb{R}^{m}$. Note for the
reader: in this section (and occasionally in Section 9) we will allow $m=0$,
but in all other sections we will assume $m\geq 1$. See also Example 3.6 and
Lemma 9.4.

######  Lemma 4.1.

There are constants $\epsilon_{mn}(n)$, $c_{mn}(n)$, so that if

$\Omega\in\mathcal{D}_{\epsilon}({\Omega^{(0)}}),\quad
M\in\mathcal{IVT}_{n}(\Omega,B_{1}),\quad||H^{tan}_{M}||_{L^{\infty}(B_{1};\mu_{M})}\leq\epsilon,$

for $\epsilon\leq\epsilon_{mn}$, then

$(1+c_{mn}(n)\epsilon\rho)^{n+1}\theta_{M}(0,\rho)$ (4.1)

is increasing in $\rho<1$. If $\epsilon=0$, and $\sigma<\rho$, then we have
the sharp monotonicity

$\theta_{M}(0,\rho)-\theta_{M}(0,\sigma)=\int_{B_{\rho}(0)\setminus
B_{\sigma}(0)}\frac{|\pi_{M}^{\perp}(z)|^{2}}{|z|^{n+2}}d\mu_{M}(z).$ (4.2)

###### Proof.

Define $Y(x)=D\Phi|_{\Phi^{-1}(x)}(\Phi^{-1}(x))$. Then $Y$ is a $C^{1}$
vector field tangential to $\Omega$. An easy computation, using (2.9) and
taking $\epsilon_{mn}(n)$ small, shows that

$|Y(x)-x|\leq c(n)\epsilon|x|^{2},\quad|DY|_{x}-\mathrm{Id}|\leq
c(n)\epsilon|x|.$ (4.3)

Let $\zeta$ be a smooth, decreasing approximation to $1_{(-\infty,1)}$, such
that $\mathrm{spt}\zeta\subset(-\infty,1)$. Take $\rho\in(0,1)$, plug in the
vector field $X(x)=\zeta(|x|/\rho)Y(x)$ into the first variation, and use
(4.3), to deduce

$\displaystyle\rho\frac{d}{d\rho}\int\zeta d\mu_{M}-n\int\zeta
d\mu_{M}-\rho\frac{d}{d\rho}\int\zeta|x^{\perp}|^{2}/|x|^{2}d\mu_{M}$ (4.4)
$\displaystyle\geq-c(n)\epsilon\int\zeta|x|d\mu_{M}-c(n)\epsilon\rho^{2}\frac{d}{d\rho}\int\zeta
d\mu_{M}.$ (4.5)

Notice that the last terms on both the RHS and LHS are non-positive. Setting
$I(\rho)=\int\zeta d\mu_{M}$, and discarding the last term on the left, we get

$(1+c\epsilon\rho)\rho I^{\prime}-(n-c\epsilon\rho)I\geq 0,$

where WLOG both constants $c=c(n)$ are the same. This implies

$\frac{d}{d\rho}((1+c\epsilon)^{n+1}\rho^{-n}I(\rho))\geq 0.$

Integrate in $\rho$, then take $\zeta\to 1_{(-\infty,1]}$, to obtain the
required conclusion (4.1).

If $\epsilon=0$, then in fact (4.4) is an equality (with no errors on the
right hand side). Integrating up as before, but without discarding terms,
gives (4.2). ∎

######  Corollary 4.2.

Given any $\theta\in(0,1)$, there are constants
$\epsilon_{m}({\Omega^{(0)}})$, $c_{m}({\Omega^{(0)}},\theta)$ so that if

$\Omega\in\mathcal{D}_{\epsilon_{m}}({\Omega^{(0)}}),\quad
M\in\mathcal{IVT}_{n}(\Omega,B_{1}),\quad||H^{tan}_{M}||_{L^{\infty}(B_{1};\mu_{M})}\leq\epsilon_{m},$

then for every $x\in B_{\theta}$, and $0<r<1-|x|$, we have

$\theta_{M}(x,r)\leq c_{m}\theta_{M}(0,1).$ (4.6)

###### Proof.

Let $\sigma_{n}=\theta$, and set
$\sigma_{i-1}=1/4+(3/4)\sigma_{i}\in(\sigma_{i},1)$ for $i=1,\ldots,n$. We
prove by induction on $i$ that there is a $c_{m0}(\theta,{\Omega^{(0)}})$ so
that given $x_{i}\in\partial_{i}\Omega\cap B_{\sigma_{i}}$, then we have

$\theta_{M}(x_{i},r)\leq c_{m0}^{i+1}\mu_{M}(B_{1})\quad\forall
0<r<1-|x_{i}|.$ (4.7)

This will establish (4.6) with $c_{m}=c_{m0}^{n+1}$.

Take $B({\Omega^{(0)}})$ as in Lemma 2.12, and let us ensure that
$\epsilon_{m}({\Omega^{(0)}})$ is sufficiently small so that
$\epsilon_{m}\leq\epsilon_{B}$, and $c_{B}\epsilon_{m}\leq\epsilon_{mn}$. Take
$x_{i}\in\partial_{i}\Omega\cap B_{\theta}$, and set
$d_{i}=d(x_{i},\partial_{i-1}\Omega)$, $R_{i}=\min\\{Bd_{i},1-|x_{i}|\\}$. For
$0<r\leq R_{i}$, we have by Lemma 2.12 and monotonicity (4.1) that

$\theta_{M}(x_{i},r)\leq c(n)\theta_{M}(x_{i},R_{i}).$

First assume $R_{i}=1-|x_{i}|$. Then we have

$\theta_{M}(x_{i},R_{i})\leq(1-\sigma_{i})^{-n}\mu_{M}(B_{1})\leq
c(n,\theta)\mu_{M}(B_{1}),$

and we are done. Notice that if $i=m$, then $d_{i}=\infty$, and so this proves
the inductive base case.

Assume now $R_{i}=Bd_{i}$. If $d_{i}\geq(1/4)(1-|x_{i}|)$, then we can
similarly estimate for $R_{i}\leq r\leq 1-|x_{i}|$:

$\theta_{M}(x_{i},r)\leq\left(\frac{4}{B(1-\sigma_{i})}\right)^{n}\mu_{M}(B_{1})\leq
c({\Omega^{(0)}},\theta)\mu_{M}(B_{1}),$

and we are done. Let us assume now $d_{i}\leq(1/4)(1-|x_{i}|)$. Since

$|x_{i}|+d_{i}\leq 1/4+(3/4)|x_{i}|\leq\sigma_{i-1},$

we can find an $x_{i-1}\in\partial_{i-1}\Omega\cap B_{\sigma_{i-1}}$ realizing
$d_{i}\equiv d(x_{i},\partial_{i-1}\Omega)$. Now by inductive hypothesis we
have for $R_{i}\leq r\leq 1-|x_{i-1}|-d_{i}$:

$\theta_{M}(x_{i},r)\leq(1+1/B)^{n}\theta_{M}(x_{i-1},r+d_{i})\leq
c({\Omega^{(0)}})c_{m0}^{i}\mu_{M}(B_{1}).$

On the other hand, since

$|x_{i-1}|+d_{i}\leq|x_{i}|+2d_{i}\leq 1/2+(1-1/2)\sigma_{i}\leq
1-1/c(n,\theta),$

if $r\geq 1-|x_{i-1}|-d_{i}$, then we have

$\theta_{M}(x_{i},r)\leq c(n,\theta)\mu_{M}(B_{1}).$

This proves the inductive claim (4.7), and finishes the proof of Corollary
4.2. ∎

######  Theorem 4.3.

There is an $\epsilon({\Omega^{(0)}})$ so that if

$\Omega\in\mathcal{D}_{\epsilon}({\Omega^{(0)}}),\quad
M\in\mathcal{IVT}_{n}(\Omega,B_{1}),\quad||H^{tan}_{M}||_{L^{\infty}(B_{1};\mu_{M})}\leq\epsilon,\quad\mu_{M}(B_{1})<\infty,$

then $||\delta M||$ is a Radon measure on $B_{1}$, and for any $X\in
C^{1}_{c}(B_{1},\mathbb{R}^{n+1})$ we can write

$\displaystyle\delta M(X)=-\int H^{tan}_{M}\cdot Xd\mu_{M}+\int\eta\cdot
Xd\sigma,$ (4.8)

where $\sigma\perp\mu_{M}$ is a non-negative Radon measure supported in
$\partial\Omega$; $\eta(x)\in\mathcal{N}_{T_{x}\Omega}$ for $\sigma$-a.e. $x$;
and $||\delta M||(\partial_{i}\Omega)=0$ for all $i\leq n-2$.

Moreover, we have

$\int\phi d||\delta M||\leq
c({\Omega^{(0)}})\int|H^{tan}_{M}|\phi+|\nabla\phi|d\mu_{M}$ (4.9)

for all $\phi\in C^{1}_{c}(B_{1})$ non-negative.

######  Remark 4.4.

Notice that (4.9) is scale-invariant.

We prove Theorem 4.3 inductively on $l$. The cases $l=1$, $l=2$, and $l\geq 3$
are handled separately. We shall use several times the following relation.

Let $Q$ be a $C^{2}$ closed $p$-manifold, with $p\leq n$, and suppose the
distance function $d$ to $Q$ is smooth on $B_{1}\setminus Q$. Suppose $\delta
M$ is a Radon measure in $B_{1}\setminus Q$, and we decompose $\delta M$ as in
(4.8) for $X$ supported away from $Q$. Let $\tau$ be any constant vector, and
$h\in C^{1}_{c}(B_{1})$ a compactly supported, non-negative function. Then we
have for any $\rho>0$:

$\frac{1}{\rho}\int_{B_{\rho}(Q)}(\nabla
d\cdot\tau)h-\int\min(d/\rho,1)h\eta\cdot\tau
d\sigma=-\int(H^{tan}_{M}h+\nabla h)\cdot\tau\min(d/\rho,1)d\mu_{M}.$ (4.10)

###### Proof when $l=0,1$.

When $l=0$ there is nothing to show. Take $l=1$. Assume $\epsilon(n)$ is
sufficiently small, so that the distance function to
$\partial\Omega\equiv\Phi(\Psi(\\{0\\}\times\mathbb{R}^{m}))\cap B_{1}$ is
smooth in $\Omega$. From Theorem 10.2, we get that $\delta M$ is a Radon
measure on $B_{1}$, and we can write $\delta M$ as in (4.8), where $\sigma$
satisfies our conclusions, and $\eta=\pm\nu_{\Omega}$. It suffices to verify
that $\eta=+\nu_{\Omega}$.

Let $d$ be the distance function to $\partial\Omega$, and $\tilde{d}$ the
signed distance function, so that $\tilde{d}=d$ in $\Omega$. Let $X=\phi
D\tilde{d}$, for $\phi\in C^{1}_{c}(B_{1})$ non-negative. From Theorem 10.2,
we have

$\int\eta\cdot(-\nu_{\Omega})\phi d\sigma=\int\eta\cdot D\tilde{d}\phi
d\sigma=-\Gamma_{1}(\phi|Dd|^{2})\leq 0.$

Since $\phi$ is arbitrary, we deduce $\eta\cdot\nu_{\Omega}\geq 0$ for
$\sigma$-a.e. $x$.

Therefore $\eta=\nu_{\Omega}$, and (using Theorem 10.2) we in fact have

$\int\phi
d\sigma=\Gamma_{1}(\phi|Dd|^{2})\leq\int|H^{tan}_{M}|\phi+|\nabla\phi|d\mu_{M}.$

Since $||\delta M||=|H^{tan}_{M}|\mu_{M}+\sigma$, we obtain (4.9). ∎

###### Proof when $l=2$.

In this case ${\Omega^{(0)}_{0}}\subset\mathbb{R}^{2}$ is simply a wedge, with
angle $<\pi$. For ease of notation let us write
$\partial{\Omega^{(0)}_{0}}=L_{1}\cup L_{2}$, for $L_{1},L_{2}$ rays extending
from $0$, and corresponding let us decompose

$\partial\Omega=F_{1}\cup F_{2},$

where $F_{i}=\Phi(\Psi(L_{i}\times\mathbb{R}^{m}))\cap B_{1}$, so that
$\partial_{m}\Omega=F_{1}\cap F_{2}$.

Away from $F_{1}\cap F_{2}$, the outer normal $\nu_{\Omega}$ is well-defined
on $\partial\Omega$. Write $\nu_{i}$ for the outward normal on $F_{i}$.
Observe that each $x\in F_{1}\cap F_{2}$, we have

$\mathcal{N}_{T_{x}\Omega}=\\{a_{1}\nu_{1}(x)+a_{2}\nu_{2}(x):a_{1},a_{2}\in(0,\infty)\\}.$

Write $d_{0},d_{1},d_{2}$ for the distance functions to $F_{1}\cap
F_{2},F_{1},F_{2}$ respectively. Convexity of ${\Omega^{(0)}_{0}}$ (and
$\mathrm{int}{\Omega^{(0)}_{0}}\neq\emptyset$) implies that by taking
$\epsilon({\Omega^{(0)}_{0}})$ sufficiently small, we can ensure the $d_{i}$
are smooth in $\mathrm{int}\Omega\cap B_{1}$, and that

$Dd_{i}\cdot Dd_{j}\geq 0\text{ on }\mathrm{int}\Omega\cap B_{1}.$ (4.11)

Also by convexity, taking $\epsilon({\Omega^{(0)}_{0}})$ small, we can choose
a vector $\tau\in\mathbb{R}^{2}\times\\{0^{m}\\}$ and an
$\epsilon_{0}({\Omega^{(0)}_{0}})>0$ so that

$\tau\cdot Dd_{i}\geq\epsilon_{0}>0\quad\text{ on }\mathrm{int}\Omega\cap
B_{1}.$

This of course implies that

$\tau\cdot\nu_{\Omega}\leq-\epsilon_{0}\text{ on }\partial\Omega\cap
B_{1}\setminus F_{1}\cap F_{2}.$ (4.12)

The upper bounds (4.6) imply that for every $\theta<1$, and any $x\in
F_{1}\cap F_{2}\cap B_{\theta}$, we have

$\mu_{M}(B_{\rho}(x))\leq c({\Omega^{(0)}},\theta)\mu_{M}(B_{1})\rho^{n}.$

In particular, we get

$\mu_{M}(B_{\rho}(F_{1}\cap F_{2})\cap B_{\theta})\leq
c({\Omega^{(0)}},\theta)\mu_{M}(B_{1})\rho.$ (4.13)

From the $l=1$ case, $||\delta M||$ is a Radon measure on
$B_{1}\setminus(F_{1}\cap F_{2})$. Take $\theta<1$, and $h\in
C^{1}_{c}(B_{\theta})$ non-negative. Let us apply formula (4.10) with our
choice of $\tau$, and $d_{0}$ in place of $d$, and make use of (4.11), (4.12),
(4.13) to obtain:

$\displaystyle\epsilon_{0}\int\min(d_{0}/\rho,1)hd\sigma$
$\displaystyle\leq\epsilon\int h+|\nabla
h|d\mu_{M}+\frac{1}{\rho}\int_{B_{\rho}(F_{1}\cap F_{2})}hd\mu_{M}$ (4.14)
$\displaystyle\leq
c|h|_{C^{1}}\mu_{M}(B_{\theta})+|h|_{C^{0}}\frac{1}{\rho}\mu_{M}(B_{\rho}(F_{1}\cap
F_{2})\cap B_{\theta})$ (4.15) $\displaystyle\leq c|h|_{C^{1}}.$ (4.16)

for $c=c({\Omega^{(0)}},\theta,\mu_{M}(B_{1}))$. Taking $\rho\to 0$, we get
that

$||\delta M||(B_{\theta}\setminus(F_{1}\cap F_{2}))<\infty,$

for all $\theta<1$.

We can now apply Theorem 10.4 to deduce that $||\delta M||$ is a Radon measure
on $B_{1}$, and we can decompose $\delta M$ as in (4.8), where
$\sigma\perp\mu_{M}$ is non-negative, supported in $\partial\Omega$; and for
$\sigma$-a.e. $x$, we have $|\eta(x)|=1$, and

  1. (1)

$\eta(x)=\nu_{i}(x)$ if $x\in F_{i}\setminus(F_{1}\cap F_{2})$);

  2. (2)

$\eta(x)\perp T_{x}(F_{1}\cap F_{2})$ if $x\in F_{1}\cap F_{2}$.

To verify the first half of our theorem, it will suffice to show that
$\eta(x)\cdot\nu_{i}(x)\geq 0$ when $x\in F_{1}\cap F_{2}$.

Let $\tilde{d}_{i}$ be the signed distance function to $F_{i}$, which
coincides with $d_{i}$ on $\mathrm{int}\Omega\cap B_{1}$. Take $\phi\in
C^{1}_{c}(B_{1})$, and let $X=\phi D\tilde{d}_{i}$. By Theorem 10.4 and (4.11)
we get

$\int_{F_{1}\cap F_{2}}\eta\cdot(-\nu_{i})\phi d\sigma=\int_{F_{1}\cap
F_{2}}\eta\cdot D\tilde{d}_{i}\phi d\sigma=-\Gamma_{2}(\phi Dd_{0}\cdot
Dd_{i})\leq 0.$

Since $\phi$ is arbitrary, we deduce $\eta\cdot\nu_{i}\geq 0$.

To prove the second part, we observe that by our choice of $\tau$ and
characterisation above, we have $\eta(x)\cdot\tau\leq-\epsilon_{0}$ for
$\sigma$-a.e. $x$. Therefore by the first variation,

$\displaystyle\epsilon_{0}\int\phi d\sigma\leq-\int\eta\cdot\tau\phi d\sigma$
$\displaystyle=\int-H^{tan}_{M}\cdot\tau\phi-\nabla\phi\cdot\tau d\mu_{M}$
(4.17) $\displaystyle\leq\int|H^{tan}_{M}|\phi+|\nabla\phi|d\mu_{M}.$ (4.18)

This completes the proof of case $l=2$. ∎

###### Proof when $l\geq 3$.

By convexity of ${\Omega^{(0)}_{0}}$ and since
$\mathrm{int}{\Omega^{(0)}_{0}}\neq\emptyset$, ensuring
$\epsilon({\Omega^{(0)}_{0}})$ is sufficiently small, we can find a vector
$\tau\in\mathbb{R}^{n+1}$, and $\epsilon_{0}({\Omega^{(0)}_{0}})>0$, so that

$\tau\cdot\nu\leq-\epsilon_{0}\quad\forall\nu\in\mathcal{N}_{T_{x}\Omega},\quad\forall
x\in\partial\Omega\cap B_{1}.$

Suppose by inductive hypothesis Theorem 4.3 holds for $1,2,\ldots,l-1$ in
place of $l$. Since (using Lemma 2.12) any point of
$\partial\Omega\setminus\partial_{m}\Omega$ is locally modelled on some
${\Omega^{(0)}_{0}}^{\prime}\times\mathbb{R}^{m+1}$, for
${\Omega^{(0)}_{0}}^{\prime}\subset\mathbb{R}^{l-1}$, we have by induction
that $||\delta M||$ is a Radon measure in $B_{1}\setminus\partial_{m}\Omega$,
and a have a decomposition of $\delta M$ as in (4.8) for any $X$ supported
away from $\partial_{m}\Omega$.

Ensuring $\epsilon({\Omega^{(0)}_{0}})$ is sufficiently small, we can assume
the distance function $d$ to $\partial_{m}\Omega$ is smooth in
$\mathrm{int}\Omega\cap B_{1}$. Analogous to the proof when $l=2$, the mass
bounds (4.6) imply that

$\mu_{M}(B_{\rho}(\partial_{m}\Omega)\cap B_{\theta})\leq
c({\Omega^{(0)}},\theta)\mu_{M}(B_{1})\rho^{2}$ (4.19)

for all $\theta<1$. Therefore, as before, if $h\in C^{1}_{c}(B_{\theta})$ is
non-negative, we can apply (4.10) with our choice of $\tau$, $d$ to deduce

$\epsilon_{0}\int\min(d/\rho,1)hd\sigma\leq c|h|_{C^{1}}$

independently of $\rho$, which implies $||\delta
M||(B_{\theta}\setminus\partial_{m}\Omega)<\infty$.

Let $X\in C^{1}_{c}(B_{\theta},\mathbb{R}^{n+1})$, for any $\theta<1$. Let
$\phi(x)=\zeta(d/\rho)$, for $\zeta$ a smooth, compactly supported
approximation to $1_{(-\infty,1]}$. We compute

$\displaystyle\left|\int div(\phi X)d\mu_{M}\right|$
$\displaystyle\leq\int|\phi^{\prime}||X|+|\phi||DX|d\mu_{M}$ (4.20)
$\displaystyle\leq
c|X|_{C^{1}}\frac{1}{\rho}\mu_{M}(B_{\rho}(\partial_{m}\Omega)\cap
B_{\theta})$ (4.21) $\displaystyle\leq c\rho\to 0\quad\text{ as }\rho\to 0.$
(4.22)

Therefore, since $||\delta M||\llcorner(\partial_{m}\Omega)^{C}$ is a Radon
measure on $B_{1}$, we deduce that

$\displaystyle\delta M(X)$ $\displaystyle=\lim_{\rho\to 0}\left(\delta M(\phi
X)+\delta M((1-\phi)X)\right)$ (4.23)
$\displaystyle=-\int_{B_{1}\setminus\partial_{m}\Omega}H^{tan}_{M}\cdot
X+\int_{B_{1}\setminus\partial_{m}\Omega}\eta\cdot Xd\sigma.$ (4.24)

So in fact $||\delta M||$ is a Radon measure on $B_{1}$, and $||\delta
M||(\partial_{m}\Omega)=0$, so that same decomposition (4.8) holds for all
$X$. The inequality (4.9) follows by the same computation as in the $l=2$
case. ∎

Inequality (4.9) is important for ensuring good compactness, but even more
importantly it implies a Sobolev inequality, which allows us to prove a mean
value inequality for subharmonic functions.

######  Theorem 4.5.

Let $M\in\mathcal{I}_{n}(B_{1})$ satisfy (4.9) for all non-negative $\phi\in
C^{1}_{c}(B_{1})$, and additionally assume that

$\mu_{M}(B_{1})\leq
A<\infty,\quad||H^{tan}_{M}||_{L^{\infty}(B_{1};\mu_{M})}\leq 1.$ (4.25)

Then the following holds: if $u\in C^{1}(B_{1})$ is a non-negative function,
and $a$ is a constant, such that

$\int\nabla\zeta\cdot\nabla ud\mu_{M}\leq a\int\zeta
u+|\nabla\zeta|u+\zeta|\nabla u|d\mu_{M}\quad\forall\zeta\in
C^{1}_{c}(B_{1})\text{ non-negative},$ (4.26)

then for every $\sigma<1$ we have the inequality

$\sup_{\mathrm{spt}\mu_{M}\cap B_{\sigma}}u\leq
c({\Omega^{(0)}},A,a,\sigma)\int_{B_{1}}ud\mu_{M}.$ (4.27)

###### Proof.

By the same argument as in [30] (see Appendix 11 for more details), inequality
(4.9) implies the Sobolev inequality: for all $h\in C^{1}_{c}(B_{1})$ non-
negative, then

$\sup h\leq c({\Omega^{(0)}})\int|H^{tan}_{M}|h+|\nabla h|d\mu_{M}$ (4.28)

if $n=1$, and

$\left(\int h^{n/(n-1)}d\mu_{M}\right)^{(n-1)/n}\leq
c({\Omega^{(0)}})\int|H^{tan}_{M}|h+|\nabla h|d\mu_{M}$ (4.29)

for $n\geq 2$. The bound (4.27) then follows from (4.25), (4.29) (or (4.28) if
$n=1$), and (4.26) by standard iteration methods. ∎

An immediate consequence is lower Ahlfors regularity in every ball centered on
the support of $M$.

######  Corollary 4.6.

There is an $\epsilon({\Omega^{(0)}})$ so that given

$\Omega\in\mathcal{D}_{\epsilon}(\Omega),\quad
M\in\mathcal{IVT}_{n}(\Omega,B_{1}),\quad||H^{tan}_{M}||_{L^{\infty}(B_{1};\mu_{M})}\leq\epsilon,\quad\theta_{M}(0,1)\leq
A,$

then for any $\theta<1$, $x\in\mathrm{spt}M\cap B_{\theta}$ and $0<r<1-|x|$,
we have

$\mu_{M}(B_{r}(x))\geq r^{n}/c({\Omega^{(0)}},A,\theta)$ (4.30)

###### Proof.

By Theorem 4.3 $M$ satisfies the scale-invariant inequality (4.9), and
therefore if $M^{\prime}=(\eta_{x,r})_{\sharp}M$ then $M^{\prime}$ satisfies
(4.9) also. From (4.6) and scaling we have the bounds

$\mu_{M^{\prime}}(B_{1})\leq
c(\Omega,A,\theta),\quad||H^{tan}_{M^{\prime}}||_{L^{\infty}(B_{1};\mu_{M^{\prime}})}\leq
r\epsilon\leq 1.$

Therefore we can apply Theorem 4.5 to $M^{\prime}$ with $u=1$, $\sigma=1/2$,
to get

$1\leq c(\Omega,A,\theta)\mu_{M^{\prime}}(B_{1}),$

which implies the required estimate (4.30). ∎

Another important consequence of first variation control and Theorem 4.5 are
the following $L^{\infty}$ and $W^{1,2}$ estimates, which will be important in
the blow-up argument to ensure good convergence to the linear problem.

######  Corollary 4.7.

Given any $\theta\in(0,1)$, there are constants
$\epsilon_{e}({\Omega^{(0)}})$, $c_{e}({\Omega^{(0)}},A,\theta)$, so that if

$\Omega\in\mathcal{D}_{\epsilon_{e}}({\Omega^{(0)}}),\quad
M\in\mathcal{IVT}_{n}(\Omega,B_{1}),\quad||H^{tan}_{M}||_{L^{\infty}(B_{1};\mu_{M})}\leq\epsilon_{e},\quad\theta_{M}(0,1)\leq
A,$

then given any $V\in\mathcal{P}_{T_{0}\Omega}$, and $p\in V^{\perp}$, we have
the height bound

$\sup_{z\in B_{\theta}\cap\mathrm{spt}M}d(z,p+V)^{2}\leq
c_{e}E(\Phi,M,p+V,0,1),$ (4.31)

and $W^{1,2}$ bound

$\int_{B_{\theta}}|\pi_{T_{z}M}-\pi_{V}|^{2}d\mu_{M}(z)\leq
c_{e}E(\Phi,M,p+V,0,1).$ (4.32)

###### Proof.

Since $V\in\mathcal{P}_{T_{0}\Omega}$, we can assume after rotation that
$V=\mathbb{R}^{n}\times\\{0\\}$. Write $\Omega=\Phi(\Psi({\Omega^{(0)}}))\cap
B_{1}$, and set $\tilde{e}_{n+1}=D\Phi_{\Phi^{-1}(x)}(e_{n+1})$. Then
$\tilde{e}_{n+1}$ is tangential to $\Omega$, and an easy computation gives
that, for $x\in B_{1}$ and $\epsilon_{e}(n)$ sufficiently small,

$|\tilde{e}_{n+1}-Dx_{n+1}|\leq
c(n)|D^{2}\Phi|_{C^{0}(B_{2})},\quad|D\tilde{e}_{n+1}|\leq
c(n)|D^{2}\Phi|_{C^{0}(B_{2})}.$ (4.33)

Ensure also that $\epsilon_{e}\leq\epsilon$, the constant from Theorem 4.3.

Given $\phi\in C^{1}(B_{1})$, plug the vector field $X=\phi\tilde{e}_{n+1}$
into the first variation, to obtain

$\int\nabla(x_{n+1}-p_{n+1})\cdot\nabla\phi\leq
c(n)(||H^{tan}_{M}||_{L^{\infty}(B_{1};\mu_{M})}+|D^{2}\Phi|_{C^{0}(B_{2})})\int|\phi|+|\nabla\phi|d\mu_{M}.$

Now if we assume $\phi\geq 0$, and let

$u=(x_{n+1}-p_{n+1})^{2}+||H^{tan}_{M}||_{L^{\infty}(B_{1};\mu_{M})}^{2}+|D^{2}\Phi|_{C^{0}(B_{2})}^{2},$

then we can compute

$\displaystyle\int\nabla u\cdot\nabla\phi d\mu_{M}$ $\displaystyle=\int
2\nabla(x_{n+1}-p_{n+1})\cdot\nabla((x_{n+1}-p_{n+1})\phi)-2\phi|\nabla
x_{n+1}|^{2}d\mu_{M}$ $\displaystyle\leq\int
c(n)(||H^{tan}_{M}||_{L^{\infty}}+|D^{2}\Phi|_{C^{0}(B_{2})})(|(x_{n+1}-p_{n+1})\phi|+|\nabla((x_{n+1}-p_{n+1})\phi)|)-2|\nabla
x_{n+1}|^{2}\phi d\mu_{M}$ $\displaystyle\leq c(n)\int
u\phi+u|\nabla\phi|d\mu_{M}.$

We can then use Theorem 4.5 to get the required (4.31).

To prove (4.32), we plug in the field
$X=\phi^{2}(x_{n+1}-p_{n+1})\tilde{e}_{n+1}$ into the first variation, and use
(4.33), to obtain:

$\displaystyle\int\phi^{2}|\nabla x_{n+1}|^{2}d\mu_{M}$ $\displaystyle\leq\int
c(n)(\phi^{2}+\phi|\nabla\phi|)(|\nabla
x_{n+1}|+|x_{n+1}-p_{n+1}|)(||H^{tan}_{M}||_{L^{\infty}(B_{1})}+|D^{2}\Phi|_{C^{0}(B_{1})})d\mu_{M}$
$\displaystyle\quad+2\phi\nabla\phi\cdot\nabla
x_{n+1}|x_{n+1}-p_{n+1}|d\mu_{M}.$

Rearranging, we get

$\int\phi^{2}|\nabla x_{n+1}|^{2}d\mu_{M}\leq\int
c(n)(\phi^{2}+|\nabla\phi|^{2})(||H^{tan}_{M}||^{2}_{L^{\infty}}+|D^{2}\Phi|^{2}_{C^{0}(B_{2})}+|x_{n+1}-p_{n+1}|^{2})d\mu_{M}.$

Choosing any fixed $\phi$ satisfying $\phi\equiv 1$ on $B_{\theta}$, and using
the relation

$|\nabla
x_{n+1}|^{2}|_{z}=(T_{z}M):V^{\perp}=\frac{1}{2}|\pi_{T_{z}M}-\pi_{V}|^{2},$

we deduce (4.32). ∎

The height bound (4.31) with the upper-Ahlfors regularity (4.6) imply that
mass cannot accumulate near the boundary of our domain.

######  Corollary 4.8.

Given $\tau,\theta\in(0,1)$, there is an
$\epsilon({\Omega^{(0)}},\theta,A,\tau)\leq\epsilon_{e}({\Omega^{(0)}})$ so
that under the assumptions of Corollary 4.7, if $M$ additionally satisfies

$E(\Phi,M,\Omega,p+V,0,1)\leq\epsilon^{2}$

for some $V\in\mathcal{P}_{T_{0}\Omega}$, $p\in V^{\perp}$, then we have the
non-concentration estimate

$\mu_{M}(B_{\theta}\cap B_{\tau}(\partial\Omega))\leq
c({\Omega^{(0)}},\theta,A)\tau.$ (4.34)

###### Proof.

There is no loss in assuming $\tau\leq(1-\theta)/4$. From the height bound
(4.31), ensuring $\epsilon({\Omega^{(0)}},\theta,A,\tau)$ is sufficiently
small, we have

$\mathrm{spt}M\cap B_{\theta}\cap B_{\tau}(\partial\Omega)\subset
B_{2\tau}(\partial[\Omega\cap(p+V)]).$

We can cover $B_{\theta}\cap B_{2\tau}(\partial[\Omega\cap(p+V)])$ with balls
$\\{B_{\tau}(x_{i})\\}_{i=1}^{Q}$ such that $Q\leq
c({\Omega^{(0)}})\tau^{1-n}$, and each $B_{\tau}(x_{i})\subset
B_{(1+\theta)/2}$. Therefore by the mass bounds (4.6) we get

$\mu_{M}(B_{\theta}\cap
B_{\tau}(\partial\Omega))\leq\sum_{i=1}^{Q}\mu_{M}(B_{\tau}(x_{i}))\leq
c({\Omega^{(0)}},A,\theta)\tau.\qed$

##  5\. Eigenvalues and the linearized operator

Let us fix here a polyhedral cone domain
${\Omega^{(0)}}={\Omega^{(0)}_{0}}^{l}\times\mathbb{R}^{m}\subset\mathbb{R}^{n+1}$,
with $m\geq 1$. We prove in this section appropriate decay of the linearized
problem on the $n$-dimensional planar wedge
${\Omega^{(0)}_{0}}\times\mathbb{R}^{m-1}\subset\mathbb{R}^{n}$. Recall that
if $D$ is a manifold with possible Lipschitz boundary $\partial D$, then
$\phi$ is said to be a Neumann eigenvalue of $D$ with eigenvalue $\mu$ if it
solves

$\int_{D}\nabla\phi\cdot\nabla\zeta+\mu\phi\zeta=0\quad\forall\zeta\in
W^{1,2}(D).$ (5.1)

Provided $D$ is compact and $\partial D$ Lipschitz, then standard elliptic
theory guarantees the existence of a countable sequence of Neumann eigenvalues
$0=\mu_{0}<\mu_{1}\leq\ldots\to\infty$, and corresponding eigenfunctions
$\phi_{i}\in W^{1,2}(D)$, so that the $\phi_{i}$ form an $L^{2}(D)$-ON basis.

We require the following Lichnerowicz eigenvalue bound for piecewise-convex
domains in the sphere. For smooth domains this is classical, while the result
piecewise smooth domains follows from a straightforward approximation.

######  Theorem 5.1 (Lichnerowicz eigenvalue bound).

Let $D$ be a piecewise-smooth, convex domain of $S^{n-1}\subset\mathbb{R}^{n}$
($n>1$), and let $\mu_{1}$ be the first (non-zero) Neumann eigenvaue, $\phi$
the first Neumann eigenfunction. Then $\mu_{1}\geq n-1$, and $\mu_{1}=n-1$ if
and only if the $1$-homogenous extension of $\phi$ is linear.

###### Proof.

First assume $D$ is smooth and weakly-convex (in the sense that the second
fundamental form of the boundary is $\geq 0$). Let $\mu$ be a Neumann
eigenvalue for $D$ in the sense of (5.1) and $u$ be the corresponding
eigenfunction. By standard elliptic regularity $u\in
C^{\infty}(\overline{D})$, and satisfies $\partial_{n}u=0$ along $\partial D$.
Therefore we can integrate the Bochner formula

$\frac{1}{2}\Delta|\nabla u|^{2}\geq(n-2)|\nabla u|^{2}+\frac{1}{n-1}(\Delta
u)^{2}-\mu|\nabla u|^{2},$

using the convexity of $D$ and boundary condition on $u$, to obtain

$(\mu-(n-1))\frac{n-2}{n-1}\int_{D}|\nabla
u|^{2}\geq\int_{D}|\nabla^{2}u|^{2}-\frac{1}{n-1}(\Delta u)^{2}.$

This proves $\mu\geq n-1$. If $\mu=n-1$, then we have
$\nabla_{ij}^{2}u+ug_{ij}=0$ ($g_{ij}$ being the spherical metric), which
implies that if we set $\tilde{u}(x)=|x|u(x/|x|)$ to be the $1$-homogenous
extension of $u$, then $D^{2}\tilde{u}=0$. This proves Theorem 5.1 in the case
when $D$ is smooth.

Now take $D$ to be convex and piecewise-smooth. Let $\mu$ be the first non-
zero Neumann eigenvalue of $D$, and $u$ the corresponding eigenfunction. Write
$\partial D=\partial^{r}D\cup\partial^{s}D$, where $\partial^{r}D$ is smooth.
Take any sequence $\epsilon_{i}\to 0$. We can find smooth, weakly convex
domains $D_{i}\subset D_{i+1}\subset\ldots\subset D$ such that $D_{i}=D$
outside $B_{\epsilon_{i}}(\partial^{s}D)$. Let $\mu_{i}$ be the first Neumman
non-zero eigenvalues of $D_{i}$, and $u_{i}$ the eigenfunctions.

Normalize $u$, $u_{i}$ such that
$||u||_{L^{2}(D)}=||u_{i}||_{L^{2}(D_{i})}=1$. It’s easy to check, using the
Raliegh quotient and standard Sobolev theory, that $\mu_{i}\to\mu$ and
$u_{i}\to u$ in $L^{2}(D^{\prime})$ for every $D^{\prime}\subset D$ such that
$D^{\prime}\cap\partial^{s}D=\emptyset$. In fact, by standard elliptic
regularity $u_{i}\to u$ in $C^{k}(D^{\prime})$. We deduce that

$(\mu-(n-1))\frac{n-2}{n-1}\mu\geq\int_{D\setminus\partial^{s}D}|\nabla^{2}u|^{2}-\frac{1}{n-1}(\Delta
u)^{2},$

and the rest of the proof follows as in the smooth case. ∎

For general Lipschitz cones, we have the following standard Fourier-type
expansion.

######  Lemma 5.2.

Let $D$ be a Lipschitz subdomain of $S^{n-1}\subset\mathbb{R}^{n}$ ($n\geq
2$), and let $CD$ be the cone over $D$. Let $\mu_{i},\phi_{i}$ be the Neumann
eigenvalues, eigenfunctions of $D\subset S^{n-1}$. Suppose $u\in
W^{1,2}(CD\cap B_{1})$ solves

$\int Du\cdot D\zeta=0\quad\forall\zeta\in C^{1}(CD)\cap
C^{1}_{c}(B_{1}^{n}).$ (5.2)

Then we have the expansion in $W^{1,2}(CD\cap B_{1})$:

$u(x=r\omega)=\sum_{i=0}^{\infty}a_{i}r^{\gamma_{i}}\phi_{i}(\omega),\quad\gamma_{i}=-(n-2)/2+\sqrt{((n-2)/2)^{2}+\mu_{i}}.$

Here we write $r=|x|$, and $\omega=x/|x|\in D$.

If $n=1$, and $u$ satisfies (5.2) for $CD=\mathbb{R}$ or $[0,\infty)$ (resp.),
then $u$ is linear or constant (resp.)

###### Proof.

This is standard, however for the reader’s convenience, and to emphasize that
we require no further regularity of $u$ beyond $W^{1,2}$, we give a proof in
Appendix 12. ∎

Combining Theorem 5.1 with Lemma 5.2 gives the following characterization of
the linear problem of our model.

######  Proposition 5.3.

Let $D_{0}={\Omega^{(0)}_{0}}\cap S^{l-1}$, and
$D=({\Omega^{(0)}}\times\mathbb{R}^{m-1})\cap S^{n-1}$. If $l\geq 2$ let
$\mu_{1}(D_{0})$ be the first Neumann eigenvalue of $D_{0}$, otherwise let us
define $\mu_{1}(D_{0})=2$.

Suppose $u\in W^{1,2}(({\Omega^{(0)}_{0}}\times\mathbb{R}^{m-1})\cap B_{1})$
solves

$\int_{{\Omega^{(0)}_{0}}\times\mathbb{R}^{m-1}}Du\cdot
D\zeta=0\quad\forall\zeta\in C^{1}_{c}(B_{1}^{n}).$ (5.3)

Then there is an increasing sequence $\\{\alpha_{i}\\}_{i=1}^{\infty}$, and
constants $b\in\mathbb{R}$, $A\in\\{0^{l}\\}\times\mathbb{R}^{m-1}$,
$a_{i}\in\mathbb{R}$, and $L^{2}(D)$-orthonormal Neumann eigenfunctions
$\psi_{i}$, so that we have the expansion in
$W^{1,2}(({\Omega^{(0)}_{0}}\times\mathbb{R}^{m-1})\cap B_{1})$:

$u(x=r\omega)=b+A\cdot x+\sum_{\alpha_{i}\geq
e}^{\infty}a_{i}r^{\alpha_{i}}\psi_{i}(\omega),$

which for every fixed $r$ is $L^{2}(D)$-orthogonal, and where

$e\geq\min\\{-(l-2)/2+\sqrt{((l-2)/2)^{2}+\mu_{1}(D_{0})},2\\}>1.$

######  Remark 5.4.

When $l=0,1$, so ${\Omega^{(0)}}$ is $\mathbb{R}^{n}$ or a half-space, then
$|\alpha_{i}|\geq 2$. When $l=2$, in which case ${\Omega^{(0)}}$ is a wedge
formed by two hyperplanes, then $|\alpha_{i}|\geq\min\\{2,\pi/\gamma\\}$,
where $\gamma$ is the angle of the wedge.

###### Proof.

If $l=0$ then $u$ is harmonic in the entire ball $B_{1}$, and then Proposition
5.3 follows by the usual Fourier expansion. Consider now $l\geq 1$. Let

$u(x=r\omega)=\sum_{i=0}^{\infty}a_{i}r^{\gamma_{i}}\phi_{i}(\omega)=:\sum_{i=0}^{\infty}u_{i}(x)$

be the expansion of Lemma 5.2. It suffices to show that if $\gamma_{i}<e$,
then $u_{i}$ is either constant, or of the form $A\cdot x$ for some
$A\in\\{0^{l}\\}\times\mathbb{R}^{m-1}$.

We have $\mu_{0}=\gamma_{0}=0$, and $\phi_{0}=\mathrm{const}$, so the first
term $u_{0}$ is a constant. Since $D$ is convex, $\gamma_{1}\geq 1$.

Suppose $\gamma_{1}=1$, so that $u_{1}$ is $1$-homogenous. Given
$v\in\\{0^{l}\\}\times\mathbb{R}^{m-1}$, then $v\cdot Du_{1}$ is a
$0$-homogenous solution to (5.3), and hence by the previous paragraph must be
constant. We deduce that $u_{1}=A\cdot x+\tilde{u}_{1}(x)$, for some
$A\in\\{0^{l}\\}\times\mathbb{R}^{m-1}$, and some $1$-homogenous
$\tilde{u}_{1}(x):{\Omega^{(0)}_{0}}\cap B_{1}\to\mathbb{R}$ solving

$\int_{{\Omega^{(0)}_{0}}\cap B_{1}}Du_{i}\cdot D\zeta=0\quad\forall\zeta\in
C^{1}_{c}(B_{1}^{l}).$ (5.4)

However, since $D_{0}$ is $0$-symmetric, by Lemma 5.2 there are no
$1$-homogenous solutions to (5.4), and so $\tilde{u}_{1}=0$.

Suppose $\gamma_{i}\in(1,2)$. Then for each $v$ as above, $v\cdot Du_{i}$ is a
$\gamma_{i}-1\in(0,1)$-homogenous solution to (5.3), and hence by the previous
two paragraphs must be zero. We deduce
$u_{i}(x)=u_{i}(\pi_{\mathbb{R}^{l}\times\\{0^{m-1}\\}}(x))$ is a function of
$\mathbb{R}^{l}$ only, and hence solves (5.4). If $l=1$ then $u_{i}$ must be
constant, which is impossible since $\gamma_{i}>1$. If $l\geq 2$, then
applying Lemma 5.2 to $u_{i}$, we deduce

$\gamma_{i}\geq-(l-2)/2+\sqrt{((l-2)/2)^{2}+\mu_{1}(D_{0})}.\qed$

##  6\. $L^{2}$ excess decay

In this section we work towards an excess decay theorem (Proposition 6.1),
which gives a decay estimate of the $L^{2}$ excess $E$ when $M$ looks close to
planar, and $\Omega$ looks close to a cone. At a general point $x$, for most
scales $\Omega$ will look like one of only finitely many cones, and so in the
next Section we will be able to use Corollary 6.2 on each model cone to prove
decay on all scales. As before, in this Section we fix ${\Omega^{(0)}}$ a
polyhedral cone domain of the form
${\Omega^{(0)}_{0}}^{l}\times\mathbb{R}^{m}$, where we assume $m\geq 1$.

######  Proposition 6.1.

Let $e({\Omega^{(0)}_{0}})$ be the exponent bound of Proposition 5.3. For any
$\theta\leq 1/10$, there are constants $\delta_{0}({\Omega^{(0)}},\theta)$,
$c_{0}({\Omega^{(0)}})$, so that the following holds. Let
$\Omega=\Phi(\Psi({\Omega^{(0)}}))\cap
B_{1}\in\mathcal{D}_{\delta_{0}}({\Omega^{(0)}})$, and
$M\in\mathcal{IVT}_{n}(\Omega,B_{1})$, satisfy

$E_{\delta_{0}}(\Phi,M,p+V,0,1)\leq
E\leq\delta_{0}^{2},\quad\theta_{M}(0,1)\leq(7/4)\Theta_{T_{0}\Omega},\quad
d_{\mathrm{spt}M}(0)\leq\delta_{0},$ (6.1)

for some $p\in V^{\perp}$, $V\in\mathcal{P}_{T_{0}\Omega}$.

Then there are $V^{\prime}\in\mathcal{P}_{T_{0}\Omega}$,
$p^{\prime}\in{V^{\prime}}^{\perp}$, satsifying

$|p-p^{\prime}|+|\pi_{V}-\pi_{V^{\prime}}|\leq c_{0}E^{1/2},$ (6.2)

so that

$E_{\delta_{0}}(\Phi,M,p^{\prime}+V^{\prime},0,\theta)\leq
c_{0}\theta^{2(e-1)}E.$ (6.3)

By iterating Proposition 6.1, we obtain the direct Corollary. As a technical
aside, instead of using the monotonicity formula (4.1) to iterate Proposition
6.1, one could use Lemma 7.2.

######  Corollary 6.2.

There are constants $\delta_{1}({\Omega^{(0)}})$, $c_{1}({\Omega^{(0)}})$,
$\beta({\Omega^{(0)}})$ so that the following holds. Let
$\Omega=\Phi(\Psi({\Omega^{(0)}}))\cap
B_{1}\in\mathcal{D}_{\delta_{1}}({\Omega^{(0)}})$,
$M\in\mathcal{IVT}_{n}(\Omega,B_{1})$, and suppose that

$E(\Phi,M,p+V,0,1)\leq
E\leq\delta_{1}^{2},\quad\theta_{M}(0,1)\leq(3/2)\Theta_{T_{0}\Omega},\quad
d_{\mathrm{spt}M}(0)\leq\rho$ (6.4)

for some $p\in V^{\perp}$, $V\in\mathcal{P}_{T_{0}\Omega}$, and some
$0\leq\rho\leq 1/2$.

Then there are $V^{\prime}\in\mathcal{P}_{T_{0}\Omega}$,
$p^{\prime}\in{V^{\prime}}^{\perp}\cap B_{1}$, satisfying

$|p-p^{\prime}|+|\pi_{V}-\pi_{V^{\prime}}|\leq(c_{1}E)^{1/2},$ (6.5)

such that

$\displaystyle E^{tot}(\Phi,M,p^{\prime}+V^{\prime},0,r)\leq
c_{1}Er^{2\beta}\quad\forall\rho\leq r\leq 1/2.$ (6.6)

######  Remark 6.3.

The exact form of the upper bound $\Theta_{T_{0}\Omega}$ is flexible. One
could equally well as use $\Theta_{{\Omega^{(0)}}}$ or
$\omega_{n+1}^{-1}\mathcal{H}^{n+1}(\Omega)$, provided we take $\delta_{1}$
sufficiently small. However in the following section we will find it
convenient to use $T_{0}\Omega$.

###### Proof of Corollary 6.2.

Choose $\theta({\Omega^{(0)}})$ sufficiently small so that
$c_{0}\theta^{2(e-1)}\leq 1/2$. Ensure $\delta_{1}({\Omega^{(0)}})$ is
sufficiently small so that

$\delta_{1}\leq\min\\{\epsilon_{mn},\epsilon_{e}\\},\quad(3/2)(1+c_{mn}\delta_{1})^{n+1}\leq
7/4\quad c_{e}\delta_{1}\leq 1/4,$

and $\delta_{1}\leq\delta_{0}^{2}$ with our choice of $\theta$. Here
$\epsilon_{mn}(n),c_{mn}(n)$ are the constants from Lemma 4.1,
$c_{e}({\Omega^{(0)}},\theta=1/2,A=2),\epsilon_{e}({\Omega^{(0)}})$ are from
Corollary 4.7, and $\delta_{0}({\Omega^{(0)}},\theta),c_{0}({\Omega^{(0)}})$
are from Proposition 6.1.

First suppose $\rho>\delta_{1}$. Then take $p^{\prime}=p$, $V^{\prime}=V$, and
we have trivially

$E(\Phi,M,p^{\prime}+V^{\prime},0,r)\leq\delta_{1}^{-n-4}r^{2\alpha}E\quad\forall\rho\leq
r\leq 1.$

Provided we take $c_{1}\geq\delta_{1}^{-n-4}$, this proves the Theorem.

Let us therefore assume $\rho\leq\delta_{1}$. Define $r_{i}=\theta^{i}$, and
let $I$ be the maximal non-negative integer for which $\rho\leq
r_{I}\delta_{1}$. If $\rho=0$, then we set $I=\infty\equiv I+1$. For
$i=0,1,\ldots,I+1$, we define inductively a sequence $p_{i}\in B_{1}$,
$V_{i}\in\mathcal{P}_{T_{0}\Omega}$, so that

$r_{i}^{-1}|p_{i+1}-p_{i}|+|\pi_{V_{i+1}}-\pi_{V_{i}}|\leq
c_{0}2^{-i/2}\delta_{0}^{-1}E^{1/2},$ (6.7)

and

$E_{\delta_{0}}(\Phi,M,p_{i}+V_{i},0,r_{i})\leq 2^{-i}\delta_{0}^{-2}E.$ (6.8)

We take $p_{0}=p$, $V_{0}=V$. Suppose, by inductive hypothesis, we have
constructed $p_{i}$, $V_{i}$ satisfying (6.8) and (if $i\geq 1$) (6.7). By our
choice of $I$ and $\delta_{1}$, and Remark 2.13, and monotonicity (4.1) we can
apply Proposition 6.1 to the rescaled varifold $(\eta_{0,r_{i}})_{\sharp}M$ in
$\Omega_{0,r_{i}}$ to obtain a $\tilde{p}_{i+1}$, $\tilde{V}_{i+1}$. Setting
$p_{i+1}=r_{i}\tilde{p}_{i+1}$ and $V_{i+1}=\tilde{V}_{i+1}$, the required
estimates (6.7), (6.8) hold by scaling. This proves the inductive step, and
therefore the existence of the required $p_{i}$, $V_{i}$.

If $I<\infty$, then set $p^{\prime}=p_{I}$, $V^{\prime}=V_{I}$. Otherwise, if
$I=\infty$, then observe that (6.7) implies the $p_{i}$, $V_{i}$ form Cauchy
sequences, and so we can take $p^{\prime}=\lim_{i\to\infty}p_{i}$,
$V^{\prime}=\lim_{i\to\infty}V_{i}$. For each (finite) $i\leq I$, we have

$r_{i}^{-1}|p^{\prime}-p_{i}|+|\pi_{V^{\prime}}-\pi_{V_{i}}|\leq
c({\Omega^{(0)}})2^{-i/2}E^{1/2}.$ (6.9)

In particular taking $i=0$ gives (6.5).

Given $\rho<r\leq 1$, either $r\geq\delta_{1}$, in which case set $i=0$, or we
have a maximal $i\leq I$ for which $r\leq\delta_{1}r_{i}$. Using (6.9) and
(4.6), we compute

$\displaystyle E(\Phi,M,p^{\prime}+V^{\prime},0,r)$
$\displaystyle\leq(\theta\delta_{1})^{-n-2}E(\Phi,M,p^{\prime}+V^{\prime},0,r_{i})$
$\displaystyle\leq
cE(\Phi,M,p_{i}+V_{i},0,r_{i})+c|\pi_{V^{\prime}}-\pi_{V_{i}}|^{2}+cr_{i}^{-2}|p^{\prime}-p_{i}|^{2}$
$\displaystyle\leq cE2^{-i}$ $\displaystyle=cEr_{i}^{2\beta}$
$\displaystyle\leq c({\Omega^{(0)}})Er^{2\alpha},$

where $\beta=\log(1/2)/\log(\theta)$. This proves the required $L^{2}$ decay
of (6.6), for $\rho\leq r\leq 1$. To deduce the decay of $E^{tot}$ we use
Remark 2.13 to apply Corollary 4.7 at each scale $\rho\leq r\leq 1/2$. ∎

The rest of the section is devoted to proving Proposition 6.1. We first
require a definition and some helper theorems.

######  Proposition 6.4 (Fine graphical approximation).

Given any $\tau,\beta$, there is a $\delta(\tau,\beta,{\Omega^{(0)}})$ and
$c({\Omega^{(0)}})$ so that the following holds. Let
$\Omega=\Phi(\Psi({\Omega^{(0)}}))\cap
B_{1}\in\mathcal{D}_{\delta}({\Omega^{(0)}})$,
$M\in\mathcal{IVT}_{n}(\Omega,B_{1})$, satisfy

$E(\Phi,M,p+V,0,1)\leq
E\leq\delta^{2},\quad\theta_{M}(0,1)\leq(7/4)\Theta_{T_{0}\Omega},\quad
d_{\mathrm{spt}M}(0)\leq\delta,$ (6.10)

for some $p\in V^{\perp}$ and $V\in\mathcal{P}_{T_{0}\Omega}$. Then the
following holds:

  1. (1)

there is a domain $U\subset p+V$, and $C^{1}$ function $u:U\to V^{\perp}$,
satisfying

$(p+V)\cap\Omega\cap B_{1/4}\setminus B_{2\tau}(\partial\Omega)\subset U,\quad
r^{-1}|u|+|Du|\leq\beta,$ (6.11)

and

$M\llcorner B_{1/2}\setminus B_{\tau}(\partial\Omega)=[\mathrm{graph}(u)];$
(6.12)

  2. (2)

we have the estimates

$\sup_{U}|u|^{2}+\int_{U}|Du|^{2}\leq cE,$ (6.13)

and

$\sup_{z\in\mathrm{spt}\mu_{M}\cap B_{1/2}}d(z,p+V)^{2}\leq cE;$ (6.14)

  3. (3)

for any $\zeta\in C^{1}_{c}(p+V,V^{\perp})\cap C^{1}_{c}(B_{1/4})$, we have

$\left|\int_{U}Du\cdot D\zeta\right|\leq
c|\zeta|_{C^{1}}\left(||H||_{L^{\infty}(B_{1};\mu_{M})}+|D^{2}\Phi|_{C^{0}(B_{2})}+\tau^{1/2}E^{1/2}\right).$

###### Proof.

Suppose, towards a contradiction, there is a sequence $\delta_{i}\to 0$,
$p_{i}\in V_{i}^{\perp}$, $V_{i}\in\mathcal{P}_{T_{0}\Omega_{i}}$,
$\Omega_{i}=\Phi_{i}(\Psi_{i}({\Omega^{(0)}}))\cap
B_{1}\in\mathcal{D}_{\delta_{i}}({\Omega^{(0)}})$, and
$M_{i}\in\mathcal{IVT}_{n}(\Omega_{i},B_{1})$, satisfying (6.10), but failing
conclusion 1. By the height bound (4.31) and our assumption
$\mathrm{spt}M_{i}\cap B_{\delta_{i}}\neq\emptyset$, we have
$d(p_{i}+V_{i},0)\to 0$, and by Theorem 4.3 we have that $||\delta M_{i}||$ is
uniformly bounded on compact subsets of $B_{1}$. We can therefore find a
$V\in\mathcal{P}_{{\Omega^{(0)}}}$, and an $M\in\mathcal{IV}_{n}(B_{1})$, so
that after passing to a subsequence, we get $p_{i}\to 0$, $V_{i}\to V$, and
$M_{i}\to M$ on compact subsets of $B_{1}$. Since
$|\Phi_{i}-\mathrm{Id}|_{C^{2}(B_{2})}\to 0$, $|\Psi_{i}-\mathrm{Id}|\to 0$,
we have $\Theta_{T_{0}\Omega_{i}}\to\Theta_{T_{0}{\Omega^{(0)}}}$.

Since $E(\Phi_{i},M,p_{i}+V_{i},0,1)\to 0$, we have $\mathrm{spt}M\subset p+V$
and $||\delta M||(\mathrm{int}{\Omega^{(0)}}\cap B_{1})=0$. Therefore by the
constancy theorem
$M\llcorner\mathrm{int}{\Omega^{(0)}}=k[p+V]\llcorner(\mathrm{int}{\Omega^{(0)}}\cap
B_{1})$ for some constant $k$. By the non-concentration estimate (4.34) we
have $\mu_{M}(\partial{\Omega^{(0)}})=0$, and trivially
$\mathrm{spt}\mu_{M}\subset{\Omega^{(0)}}$, and so in fact
$M=k[p+V]\llcorner(\mathrm{int}{\Omega^{(0)}}\cap B_{1})$. From lower-
regularity (4.30), $\mu_{M_{i}}(B_{1/2})\geq 1/c({\Omega^{(0)}})$, and hence
$\mu_{M}(B_{1})\geq 1/c({\Omega^{(0)}})$. This implies $k\geq 1$. On the other
hand,

$\mu_{M}(B_{1})\leq\liminf_{i}\mu_{M_{i}}(B_{1})\leq(3/2)\lim_{i}\Theta_{T_{0}\Omega_{i}}=(3/2)\Theta_{T_{0}{\Omega^{(0)}}},$

and so $k\leq 1$. We deduce $k=1$. Since for every
$W\subset\subset\mathrm{int}{\Omega^{(0)}}$ and $i>>1$ the $M_{i}\llcorner W$
have bounded mean curvature (tending to zero) and zero boundary, Allard’s
theorem [1] implies the convergence is $C^{1}$ on compact subsets of
$\mathrm{int}{\Omega^{(0)}}\cap B_{1}$. This proves conclusion 1.

Let us now fix a $\tau,U$, and prove conclusions 2 and 3. Ensuring
$\delta({\Omega^{(0)}})$ is sufficiently small, There is no loss in assuming
$p+V=\mathbb{R}^{n}\times\\{0\\}$. Given a function $f:D\subset(p+V)\to
V^{\perp}$, we can extend $f$ to be defined in $D\times V^{\perp}$ by setting
$\tilde{f}(x)=f(p+\pi_{V}(x-p))$. Given $f,g:D\to V^{\perp}$, then at
$\mu_{M}$-a.e. $x\in\mathrm{spt}M\cap(D\times V^{\perp})$ we have the bound

$|\nabla\tilde{f}\cdot\nabla\tilde{g}-D\tilde{f}\cdot
D\tilde{g}|\leq|\pi_{T_{x}M}-\pi_{V}|^{2}|D\tilde{f}||D\tilde{g}|.$ (6.15)

In the special case when $f=u$ as defined on $U$, then

$|\nabla\tilde{u}|^{2}=|\nabla x_{n+1}|^{2}\leq|\pi_{T_{x}M}-\pi_{V}|^{2}.$
(6.16)

Combining (6.15), (6.16) with (4.32), and (without loss of generality)
ensuring $\beta\leq\beta(n)$, we get

$\int_{U}|Du|^{2}\leq(1+c(n)\beta^{2})\int_{B_{1/2}}|D\tilde{u}|^{2}d\mu_{M}\leq
c(n)\int_{B_{1/2}}|\pi_{T_{z}M}-\pi_{V}|^{2}d\mu_{M}(z)\leq
c({\Omega^{(0)}})E.$

This completes the $W^{1,2}$ estimate of conclusion 2; the $L^{\infty}$
estimates both follow from the sharp height bound (4.31).

We prove conclusion 3. Take $\zeta\in C^{1}_{c}(p+V,V^{\perp})\cap
C^{1}_{c}(B_{1/4})$. Let $\phi$ be any function which is $1$ on $B_{1/2}$, and
is supported in $B_{1}$. By the height bound (4.31) we can assume that
$\mathrm{spt}M\cap\mathrm{spt}\tilde{\zeta}\subset B_{1/2}$. Plugging the
field $X=\tilde{\zeta}\phi\tilde{e}_{n+1}$ into the first variation (where
$\tilde{e}_{n+1}$ as in the proof of Corollary 4.7), and noting that $X$ is
tangential to $\Omega$, we obtain

$\left|\int_{B_{1/2}}\nabla\tilde{\zeta}\cdot\nabla x_{n+1}d\mu_{M}\right|\leq
c(n)|\zeta|_{C^{1}}(||H^{tan}_{M}||_{L^{\infty}(B_{1};\mu_{M})}+|D^{2}\Phi|_{C^{0}(B_{2})}).$
(6.17)

Let us make some remarks. First, provided $\beta(n)$ is sufficiently small,
the Jacobian $Ju$ of $u$ satisfies the bounds

$1\leq Ju\leq 1+c(n)|Du|^{2}.$ (6.18)

Second, ensuring $\delta(\tau,{\Omega^{(0)}})$ is sufficiently small, by the
non-concentration estimate (4.34) we have

$\mu_{M}(B_{\tau}(\partial\Omega)\cap B_{1/2})\leq c({\Omega^{(0)}})\tau.$

Last, using (4.32), this implies that

$\displaystyle\left|\int_{B_{1/2}\cap
B_{\tau}(\partial\Omega)}\nabla\tilde{\zeta}\cdot e_{n+1}d\mu_{M}\right|$
$\displaystyle\leq
c({\Omega^{(0)}})|\zeta|_{C^{1}}\tau^{1/2}\left(\int_{B_{1/2}}|\pi_{T_{z}M}-\pi_{V}|^{2}d\mu_{M}(z)\right)^{1/2}$
$\displaystyle\leq c({\Omega^{(0)}})|\zeta|_{C^{1}}\tau^{1/2}E^{1/2}$ (6.19)

Using conclusions 1 and 2, equations (6.17), (6.18), (6.19) we obtain

$\displaystyle\left|\int_{U}D\zeta\cdot Du\right|$ $\displaystyle\leq
c\beta|\zeta|_{C^{1}}\int_{U}|Du|^{2}+\left|\int_{B_{1/2}\setminus
B_{\tau}(\partial\Omega)}D\tilde{\zeta}\cdot D\tilde{u}d\mu_{M}\right|$
$\displaystyle\leq
c|\zeta|_{C^{1}}\int_{U}|Du|^{2}+c|\zeta|_{C^{1}}\int_{B_{1/2}}|\pi_{T_{z}M}-\pi_{V}|^{2}d\mu_{M}(z)+\left|\int_{B_{1/2}\setminus
B_{\tau}(\partial\Omega)}\nabla\tilde{\zeta}\cdot e_{n+1}d\mu_{M}\right|$
$\displaystyle\leq
c|\zeta|_{C^{1}}E+c|\zeta|_{C^{1}}\tau^{1/2}E^{1/2}+c|\zeta|_{C^{1}}(||H||_{L^{\infty}(B_{1};\mu_{M})}+|D^{2}\Phi|_{C^{0}(B_{2})})$
$\displaystyle\leq
c|\zeta|_{C^{1}}\tau^{1/2}E^{1/2}+c|\zeta|_{C^{1}}(||H||_{L^{\infty}(B_{1};\mu_{M})}+|D^{2}\Phi|_{C^{0}(B_{2})}),$

for $c=c({\Omega^{(0)}})$. The last inequality follows because we can of
course assume $\delta^{2}\leq\tau$. This completes conclusion 3, and the proof
of Proposition 6.4. ∎

######  Definition 6.4.1.

Consider the sequences $\beta_{i},\delta_{i}\in\mathbb{R}$,
$\Omega_{i}=\Phi_{i}(\Psi_{i}({\Omega^{(0)}}))\cap
B_{1}\in\mathcal{D}_{\delta_{i}}({\Omega^{(0)}})$,
$V_{i}\in\mathcal{P}_{T_{0}\Omega_{i}}$, $p_{i}\in V_{i}^{\perp}$,
$M_{i}\in\mathcal{IVT}_{n}(\Omega,B_{1})$. We say
$({\Omega^{(0)}},\Omega_{i},M_{i},p_{i}+V_{i},\beta_{i},\delta_{i})$ is a
blow-up sequence if:

  1. (1)

$p_{i}\to 0$, $V_{i}\to\mathbb{R}^{n}\times\\{0\\}$, $\beta_{i}\to 0$,
$\delta_{i}\to 0$,

  2. (2)

$\mu_{M_{i}}(B_{1})\leq(7/4)\Theta_{T_{0}\Omega_{i}}$, and
$d_{\mathrm{spt}M_{i}}(0)\to 0$,

  3. (3)

$\limsup_{i}\beta_{i}^{-2}E_{\delta_{i}}(\Phi_{i},M_{i},p_{i}+V_{i},0,1)<\infty$.

######  Proposition 6.5.

Let $({\Omega^{(0)}},\Omega_{i},M_{i},p_{i}+V_{i},\beta_{i},\delta_{i})$ be a
blow-up sequence. Let us write
$p_{i}+V_{i}=\mathrm{graph}_{\mathbb{R}^{n}\times\\{0\\}}(q_{i}+\phi_{i}\cdot
x)$, for $q_{i}\in\mathbb{R}$ and
$\phi_{i}\in\\{0^{l}\\}\times\mathbb{R}^{m-1}$. Then there is a $W^{1,2}$
function $v:({\Omega^{(0)}_{0}}\times\mathbb{R}^{m-1})\cap
B_{1/4}\to\mathbb{R}$, such that:

  1. (1)

$v$ is weakly harmonic with Neumann boundary:

$\int_{({\Omega^{(0)}_{0}}\times\mathbb{R}^{m-1})\cap B_{1/4}}Dv\cdot
D\zeta=0\quad\forall\zeta\in
C^{1}_{c}(B_{1/4}\cap(\mathbb{R}^{n}\times\\{0\\}));$ (6.20)

  2. (2)

$v$ has the $W^{1,2}$ bound

$\int_{({\Omega^{(0)}_{0}}\times\mathbb{R}^{m-1})\cap
B_{1/4}}v^{2}+|Dv|^{2}\leq
c(\Omega)\left(\limsup_{i}\beta_{i}^{-2}E_{\delta_{i}}(\Phi_{i},M_{i},p_{i}+V_{i},0,1)\right);$
(6.21)

  3. (3)

on any compact set $U\subset\subset
B_{1/4}\cap({\Omega^{(0)}_{0}}\times\mathbb{R}^{m-1})\setminus\partial{\Omega^{(0)}}$,
we have $L^{2}$ convergence

$\beta_{i}^{-1}u_{i}(x+q_{i}+\phi_{i}\cdot x)\to v(x);$ (6.22)

  4. (4)

for any $\rho\leq 1/4$, we have the $L^{2}$ convergence

$\beta_{i}^{-2}\int_{B_{\rho}}d_{p_{i}+V_{i}}^{2}d\mu_{M}\to\int_{B_{\rho}\cap({\Omega^{(0)}_{0}}\times\mathbb{R}^{m-1})}v^{2}.$
(6.23)

###### Proof.

Let $\tau_{i}\to 0$ sufficiently slowly, so that for each $i$ large we can
apply Proposition 6.4 to to deduce

$M_{i}\cap B_{1/2}\setminus
B_{\tau_{i}}(\partial{\Omega^{(0)}})=\mathrm{graph}_{p_{i}+V_{i}}(u_{i}),$

where $u_{i}$ is defined on some domain $U_{i}$ satisfying

$(p_{i}+V_{i})\cap B_{1/4}\setminus
B_{2\tau_{i}}(\partial{\Omega^{(0)}})\subset U_{i}\subset p_{i}+V_{i}.$

Write $E_{i}=E_{\delta_{i}}(\Phi_{i},M_{i},p_{i}+V_{i},0,1)$.

Fix any $U\subset\subset
B_{1/4}\cap({\Omega^{(0)}_{0}}\times\mathbb{R}^{m-1})\setminus\partial{\Omega^{(0)}}$.
Then for sufficiently large $i$, $x\mapsto
w_{i}(x):=u_{i}(x+q_{i}+\phi_{i}\cdot x)$ is well-defined, and parameterizes a
subset of $\mathrm{spt}M_{i}$. For $x\in U$, we have

$Dw_{i}(x)=(1+o(1))Du_{i}(x+q_{i}+\phi_{i}\cdot x),\quad 1\leq J(x)\leq
1+o(1),$

where $J(x)$ is the Jacobian of $x\mapsto x+q_{i}+\phi_{i}\cdot x$. It follows
from Proposition 6.4 that

$\int_{U}|w_{i}|^{2}+|Dw_{i}|^{2}\leq cE_{i},\quad\sup_{U}|w_{i}|^{2}\leq
cE_{i},$ (6.24)

and for any $\zeta\in C^{1}_{c}(B_{1/4})$,

$\left|\int_{U}D\zeta\cdot Dw_{i}\right|\leq o(1)|\zeta|_{C^{1}}E_{i}^{1/2}.$
(6.25)

If $v_{i}=\beta_{i}^{-1}w_{i}$, then (6.24) (and our definition of blow-up
sequence) shows that the $v_{i}$ are bounded in $W^{1,2}(U)$, with a bound
independent of either $i$ or $U$. A diagonalization argument implies there is
a $v\in W^{1,2}(B_{1/4}\cap({\Omega^{(0)}_{0}}\times\mathbb{R}^{m-1}))$,
satisfying the bound (6.21), so that for every $U$ as above $v_{i}\to v$
strongly in $L^{2}(U)$ and weakly in $W^{1,2}(U)$.

From (6.24) we have

$\sup|v|^{2}\leq c({\Omega^{(0)}}),$ (6.26)

and from (6.25) we get that $v$ satisfies the required (6.20). The strong
$L^{2}$ convergence (6.23) follows from the $L^{\infty}$ bounds (6.26),
(6.14), (4.34) and the fact

$\int_{B_{\rho}\setminus
B_{\tau_{i}}(\partial{\Omega^{(0)}})}d_{p_{i}+V_{i}}^{2}d\mu_{M}=(1+o(1))\int_{U_{i}}|u_{i}|^{2}.\qed$

######  Definition 6.5.1.

Let us call any $v$ as obtained in Proposition 6.5 a _Jacobi field_ on
${\Omega^{(0)}_{0}}\times\mathbb{R}^{m-1}$.

We are now ready to prove Proposition 6.1.

###### Proof of Proposition 6.1.

Suppose, towards a contradiction, there is a sequence $\delta_{i}\to 0$,
$V_{i}\in\mathcal{P}_{T_{0}\Omega_{i}}$, $p_{i}\in V_{i}^{\perp}$,
$\Omega_{i}=\Phi_{i}(\Psi_{i}({\Omega^{(0)}}))\in\mathcal{D}_{\delta_{i}}({\Omega^{(0)}})$,
and $M_{i}\in\mathcal{IVT}_{n}(\Omega_{i},B_{1})$, such that

$E_{i}:=E_{\delta_{i}}(\Phi_{i},M_{i},p_{i}+V_{i},0,1)\leq\delta_{i}^{2},\quad\mu_{M_{i}}(B_{1})\leq(7/4)\Theta_{T_{0}\Omega_{i}},\quad\mathrm{spt}M_{i}\cap
B_{\delta_{i}}\neq\emptyset,$

but for which

$E_{\delta_{i}}(\Phi_{i},M_{i},p^{\prime}+V^{\prime},0,\theta)\geq
c_{0}\theta^{2(e-1)}E_{i},$

for all $p^{\prime}+V^{\prime}$ satisfying (6.2). Here $c_{0}$ is a constant
depending only on ${\Omega^{(0)}}$ that we will choose later.

Let $\beta_{i}^{2}=E_{\delta_{i}}(\Phi_{i},M_{i},p_{i}+V_{i},0,1)$, so of
course $\beta_{i}\to 0$ also. From the height bound (4.31), we can assume
$p_{i}\to 0$, and after passing to a subsequence and rotating as necessary, we
can assume $V_{i}\to\mathbb{R}^{n}\times\\{0\\}$. Then
$({\Omega^{(0)}},\Omega_{i},M_{i},p_{i}+V_{i},\beta_{i},\delta_{i})$ is a
blow-up sequence, and we can apply Proposition 6.5 to obtain a Jacobi field
$v$, satisfying

$\int_{{\Omega^{(0)}_{0}}\times\mathbb{R}^{m-1}}Dv\cdot
D\zeta=0\quad\forall\zeta\in
C^{1}_{c}(B_{1/4}\cap(\mathbb{R}^{n}\times\\{0\\})),$

and the $W^{1,2}$ estimate

$\int_{({\Omega^{(0)}_{0}}\times\mathbb{R}^{m-1})\cap
B_{1/4}}|v|^{2}+|Dv|^{2}\leq c({\Omega^{(0)}}).$

From Proposition 5.3, we can expand in $W^{1,2}$

$v(x=r\omega)=b+A\cdot x+\sum_{i}a_{i}r^{\alpha_{i}}\phi_{i}(\omega),$ (6.27)

for some $A\in\\{0^{l}\\}\times\mathbb{R}^{m-1}$, and $\alpha_{i}\geq e>1$,
for $e({\Omega^{(0)}_{0}})$ as in 5.3. Using the
$L^{2}(({\Omega^{(0)}_{0}}\times\mathbb{R}^{m-1})\cap S^{n-1})$-orthogonality
of this expansion for every fixed $r$, we get

$|b|^{2}+|A|^{2}+\sum_{i}\frac{a_{i}^{2}(1/4)^{2\alpha_{i}+n}}{2\alpha_{i}+n}\leq
c({\Omega^{(0)}}).$ (6.28)

Let us write

$p_{i}+V_{i}=\mathrm{graph}_{\mathbb{R}^{n}\times\\{0\\}}(q_{i}+\phi_{i}\cdot
x),$

for $q_{i}\in\mathbb{R}$, and $\phi_{i}\in\\{0^{l}\\}\times\mathbb{R}^{m-1}$.
Now define the new affine planes

$p_{i}^{\prime}+V_{i}^{\prime}=\mathrm{graph}_{\mathbb{R}^{n}\times\\{0\\}}(q_{i}+\beta_{i}b+(\phi_{i}+\beta_{i}A)\cdot
x).$

Since $A\in\\{0^{l}\\}\times\mathbb{R}^{m-1}$, we have
$V_{i}^{\prime}\in\mathcal{P}_{\Omega^{(0)}}$, and we can take
$p_{i}^{\prime}\in{V_{i}^{\prime}}^{\perp}$. By definition of the
$p_{i}^{\prime}$, $V_{i}^{\prime}$, and by considering the analytic maps

$\phi\mapsto\pi_{\phi}:=\pi_{\mathrm{graph}_{\mathbb{R}^{n}\times\\{0\\}}(\phi\cdot
x)},\quad(q,\phi)\mapsto\pi^{\perp}_{\phi}(qe_{n+1})$

we have

$|p_{i}-p_{i}^{\prime}|+|\pi_{V_{i}}-\pi_{V_{i}^{\prime}}|\leq
c(n)\beta_{i}(|b|+|A|)\leq c_{d1}({\Omega^{(0)}})\beta_{i}.$ (6.29)

Therefore,
$({\Omega^{(0)}},\Omega_{i},M_{i},p_{i}^{\prime}+V_{i}^{\prime},\beta_{i},\delta_{i})$
is a blow-up sequence also, and we can again use Proposition 6.5 to obtain a
new Jacobi field $v^{\prime}$ (with, a priori, a slightly worse $W^{1,2}$
bound than $v$).

There is a sequence $\tau_{i}\to 0$, so that we can write

$M_{i}\cap B_{1/2}\setminus
B_{\tau_{i}}(\partial{\Omega^{(0)}})=\mathrm{graph}_{p_{i}+V_{i}}(u_{i})=\mathrm{graph}_{p_{i}^{\prime}+V_{i}^{\prime}}(u_{i}^{\prime}),$
(6.30)

where, for any compact subset $U\subset\subset
B_{1/4}\cap(\mathrm{int}{\Omega^{(0)}_{0}}\times\mathbb{R}^{m-1})$, we have

$|u_{i}(x+q_{i}+\phi_{i}\cdot
x)|_{C^{1}(U)}+|u_{i}^{\prime}(x+q_{i}+\beta_{i}b+(\phi_{i}+\beta_{i}A)\cdot
x)|_{C^{1}(U)}\to 0,$ (6.31)

and (from Proposition 6.5.3)

$\beta_{i}^{-1}u_{i}(x+q_{i}+\phi_{i}\cdot x)\to
v,\quad\beta_{i}^{-1}u_{i}^{\prime}(x+q_{i}+\beta_{i}p+(\phi_{i}+\beta_{i}A)\cdot
x)\to v^{\prime}$ (6.32)

in $L^{2}(U)$.

From (6.30), (6.31) and since $|p_{i}|+|\phi_{i}|\to 0$, we have that for
every $x\in U$,

$|u_{i}^{\prime}(x+q_{i}+\beta_{i}p+(\phi_{i}+\beta_{i}A)\cdot
x)-u_{i}(x+q_{i}+\phi_{i}\cdot x)-\beta_{i}p-\beta_{i}A\cdot x|\leq
o(1)\beta_{i},$ (6.33)

where $o(1)\to 0$ as $i\to\infty$. By (6.32) and (6.33), we deduce that

$v^{\prime}(x)=v(x)-b-A\cdot x=\sum_{i}a_{i}r^{\alpha_{i}}\phi_{i}(x),$

where $a_{i}$ as in equation (6.27).

Therefore, using (6.28), $4\theta\leq 1$ and $|\alpha_{i}|\geq 1+\alpha$, we
have:

$\int_{B_{\theta}\cap({\Omega^{(0)}_{0}}\times\mathbb{R}^{m-1})}|v^{\prime}|^{2}=\sum_{i}\frac{a_{i}^{2}\theta^{2\alpha_{i}+n}}{2\alpha_{i}+n}\leq
c_{d2}({\Omega^{(0)}})(4\theta)^{n+2e}.$

Provided we take $c_{0}$ larger than $c_{d1}$ and $8^{4+n}c_{d2}$, then by the
strong $L^{2}$ convergence

$\beta_{i}^{-2}\int_{B_{\theta}}d_{p_{i}^{\prime}+V_{i}^{\prime}}^{2}d\mu_{M}\to\int_{B_{\theta}\cap({\Omega^{(0)}_{0}}\times\mathbb{R}^{m-1})}|v^{\prime}|^{2},$

we obtain a contradiction (we remind the reader that the $H$ and $\Phi$ terms
of $E$ decay gratuitously). This completes the proof of Proposition 6.1. ∎

##  7\. Regularity

The main theorem of this section is the following decay estimate, which we
shall prove by induction on the boundary strata $\partial_{i}\Omega$. The idea
is that we can use the decay of Proposition 6.1 (or rather Corollary 6.2)
whenever $\Omega$ resembles a polyhedral cone. If we hit a radius at which
$\Omega$ stops resembling a cone, then by recentering on a lower strata and
dropping a controllable number of scales, we will start looking like a cone
with an extra degree of symmetry.

Although the $L^{2}$, $W^{1,2}$, and $L^{\infty}$ distances to planes are all
effectively comparable when $\Omega$ resembles a cone (and the plane lies in
the “good” space $\mathcal{P}_{T_{0}\Omega}$), as we traverse scales and cone-
types it will be convenient to prove a decay on the $L^{2}$, $W^{1,2}$, and
$L^{\infty}$ excesses simultaneously. The proof is no more involved than
proving a decay on the $L^{2}$ by itself.

As in the previous sections we fix here a polyhedral cone domain
${\Omega^{(0)}}={\Omega^{(0)}_{0}}^{l}\times\mathbb{R}^{m}$, having $m\geq 1$.

######  Theorem 7.1.

There are constants $c_{3}({\Omega^{(0)}})$, $\delta_{3}({\Omega^{(0)}})$,
$\alpha({\Omega^{(0)}})$ so that the following holds. Let
$\Omega=\Phi(\Psi({\Omega^{(0)}}))\cap
B_{1}\in\mathcal{D}_{\delta_{3}}({\Omega^{(0)}})$,
$M\in\mathcal{IVT}_{n}(\Omega,B_{1})$ satisfy

$E(\Phi,M,p+V,0,1)\leq
E\leq\delta_{3}^{2},\quad\theta_{M}(0,1)\leq(3/2)\Theta_{T_{0}\Omega},$ (7.1)

for some $p\in V^{\perp}$, $V\in\mathcal{P}_{T_{0}\Omega}$.

Then for every $x\in\mathrm{spt}M\cap B_{1/16}$, there is plane
$V_{x}\in\mathcal{P}_{T_{x}\Omega}$ so that

$|\pi_{V}-\pi_{V_{x}}|\leq c_{3}E^{1/2},$ (7.2)

and for all $0<r<1/4$:

$E^{tot}(\Phi,M,x+V_{x},x,r)\leq c_{3}r^{2\alpha}E$ (7.3)

We require first a helper lemma.

######  Lemma 7.2.

Given any $\epsilon>0$, there is a $\delta_{4}({\Omega^{(0)}},\epsilon)$ so
that the following holds. Let $\Omega=\Phi(\Psi({\Omega^{(0)}}))\cap
B_{1}\in\mathcal{D}_{\delta_{4}}({\Omega^{(0)}})$,
$M\in\mathcal{IVT}_{n}(\Omega,B_{1})$ satisfy

$E(\Phi,M,p+V,0,1)\leq\delta_{4}^{2},\quad\theta_{M}(0,1)\leq(7/4)\Theta_{T_{0}\Omega},$
(7.4)

for some $p\in V^{\perp}$, and some $V\in\mathcal{P}_{T_{0}\Omega}$. Then
given any $x\in\Omega\cap B_{1}$ and $r\geq\epsilon$, so that $B_{r}(x)\subset
B_{1-\epsilon}$, we have

$\theta_{M}(x,r)\leq(3/2)\Theta_{T_{x}\Omega}.$ (7.5)

###### Proof.

Proof by contradiction. Suppose otherwise: there is a sequence $\delta_{i}\to
0$, domains $\Omega_{i}=\Phi_{i}(\Psi_{i}({\Omega^{(0)}}))\cap
B_{1}\in\mathcal{D}_{\delta_{i}}({\Omega^{(0)}})$, and varifolds
$M_{i}\in\mathcal{IVT}_{n}(\Omega_{i},B_{1})$, so that

$E(\Phi_{i},M_{i},p_{i}+V_{i},0,1)\to
0,\quad\theta_{M_{i}}(0,1)\leq(7/4)\Theta_{T_{0}\Omega_{i}},$

for some sequence $V_{i}\in\mathcal{P}_{T_{0}\Omega_{i}}$, $p_{i}\in
V_{i}^{\perp}$, but there are points $x_{i}\in\Omega_{i}\cap B_{1}$, and radii
$r_{i}\geq\epsilon$ such that

$B_{r_{i}}(x_{i})\subset
B_{1-\epsilon},\quad\theta_{M_{i}}(x_{i},r_{i})\geq(3/2)\Theta_{T_{x_{i}}\Omega_{i}}.$

By the height bound (4.31) there is no loss in assuming the $p_{i}$ are
bounded, as otherwise we would have $\mathrm{spt}M_{i}\cap B_{1}=\emptyset$
for large $i$. We can therefore pass to a subsequence, and get convergence
$p_{i}\to p^{\prime}\in\mathbb{R}^{n+1}$, $V_{i}\to
V^{\prime}\in\mathcal{P}_{T_{0}{\Omega^{(0)}}}$, $x_{i}\to x\in B_{1}$,
$r_{i}\to r\geq\epsilon$, so that $B_{r}(x)\subset B_{1-\epsilon}$, and and
$M_{i}\to M^{\prime}$, for some $M^{\prime}\in\mathcal{IV}_{n}(B_{1})$. As in
the proof of Proposition 6.4, we must have
$M^{\prime}=[p^{\prime}+V^{\prime}]\llcorner(\mathrm{int}{\Omega^{(0)}}\cap
B_{1})$.

For a.e. $1-|x|>\rho>r$ we have by the lower-semi-continuity (2.10):

$\mu_{M}(B_{\rho}(x))=\lim_{i}\mu_{M_{i}}(B_{\rho}(x))\geq\limsup_{i}\mu_{M_{i}}(B_{r_{i}}(x_{i}))\geq(3/2)\Theta_{T_{x}{\Omega^{(0)}}}\omega_{n}r^{n}.$

On the other hand, since $V\in\mathcal{P}_{\Omega^{(0)}}$, we can use
monotonicity (2.6) to get

$\mu_{M}(B_{\rho}(x))=\mathcal{H}^{n}((p+V)\cap{\Omega^{(0)}}\cap
B_{\rho}(x))\leq\mathcal{H}^{n}((x+V)\cap{\Omega^{(0)}}\cap
B_{\rho}(x))\leq\Theta_{T_{x}{\Omega^{(0)}}}\omega_{n}\rho^{n},$

which is a contradiction for $\rho$ sufficiently close to $r$. ∎

###### Proof of Theorem 7.1.

There is no loss in assuming $p\in B_{1}$. Let $B({\Omega^{(0)}})$ be as in
Lemma 2.12. Observe that since the set $\\{T_{z}{\Omega^{(0)}}\\}_{z}$ is
finite, any constant that depends on some $T_{z}{\Omega^{(0)}}$ can be made to
depend only on ${\Omega^{(0)}}$. In particular, let us choose $\alpha$ by
setting

$\alpha=\min\\{\beta(T_{z}{\Omega^{(0)}})\\}_{z}$

where $\beta$ as in Corollary 6.2. In the following $c$ will denote a generic
constant $\geq 1$ depending only on ${\Omega^{(0)}}$, which may increase from
line to line.

Take $x\in\partial_{m+i}\Omega\cap B_{1/16}$ (note $i\geq 0$). Let $x_{i}=x$,
and then for $j=i,\ldots,1$ define $x_{j-1}\in\partial_{m+j-1}\Omega$ to be
the point realizing $d(x_{j},\partial_{m+j-1}\Omega)$. Let us formally define
$(B/2)|x_{0}-x_{-1}|=1/8$.

Let $i_{0}=i$. If $i=0$ let $J=0$. Otherwise, define inductively
$j=0,1,2,\ldots,J$ by the conditions that $0\leq i_{j+1}<i_{j}$, and

$|x_{i_{j}}-x_{i_{j+1}}|+B|x_{i_{j}}-x_{i_{j}-1}|\leq(B/2)|x_{i_{j+1}}-x_{i_{j+1}-1}|,$

but

$|x_{i_{j}}-x_{k}|+B|x_{i_{j}}-x_{i_{j}-1}|>(B/2)|x_{k}-x_{k-1}|\quad\forall
k=i_{j+1}+1,\ldots,i_{j}-1.$

Note that since $|x|\leq 1/16$, $0\in\partial_{m}\Omega$, and by Lemma 2.12(1)
we have (taking $\delta_{3}(n)$ small)

$|x_{i_{j}}-x_{k}|+B|x_{i_{j}}-x_{i_{j}-1}|\leq(3/2)|x_{i_{j}}|\leq(3/2)(1+c_{B}\delta_{3})|x|\leq
1/8,$

for every $k\leq i_{j}-1$, and in particular we have $i_{J}=0$.

Let $r_{0}^{-}=0$. Define

$\displaystyle r_{j}^{+}$
$\displaystyle=B|x_{i_{j}}-x_{i_{j}-1}|\quad(j=0,\ldots,J),$ $\displaystyle
r_{j}^{-}$
$\displaystyle=2r^{+}_{j-1}+2|x_{i_{j-1}}-x_{i_{j}}|\quad(j=1,\ldots,J).$

Note that since $i_{J}=0$, we have $r_{J}^{+}=1/4$. By construction we have
the inclusions

$x\in B_{r_{j-1}^{-}}(x_{i_{j-1}})\subset B_{r_{j-1}^{+}}(x_{i_{j-1}})\subset
B_{r_{j}^{-}/2}(x_{i_{j}})\subset B_{1}(0).$ (7.6)

Now for $j=0,1,\ldots,J-1$ we have

$\displaystyle|x_{i_{j}}-x_{i_{j+1}}|$
$\displaystyle\leq|x_{i_{j}}-x_{i_{j+1}+1}|+|x_{i_{j+1}+1}-x_{i_{j+1}}|$
$\displaystyle\leq|x_{i_{j}}-x_{i_{j+1}+1}|+(2/B)(|x_{i_{j}}-x_{i_{j+1}+1}|+B|x_{i_{j}}-x_{i_{j}-1}|)$
$\displaystyle\leq 2|x_{i_{j}}-x_{i_{j}-1}|+(1+2/B)|x_{i_{j}}-x_{i_{j+1}+1}|$
$\displaystyle\leq
2(1+(1+2/B))|x_{i_{j}}-x_{i_{j}-1}|+(1+2/B)^{2}|x_{i_{j}}-x_{i_{j+1}+2}|$
$\displaystyle\leq\ldots$ $\displaystyle\leq c(n,B)|x_{i_{j}}-x_{i_{j}-1}|.$

Therefore

$\displaystyle\frac{r_{j}^{+}}{r_{j+1}^{-}}$
$\displaystyle=\frac{B|x_{i_{j}}-x_{i_{j}-1}|}{2|x_{i_{j}}-x_{i_{j+1}}|+2B|x_{i_{j}}-x_{i_{j}-1}|}$
$\displaystyle=\frac{1}{(2/B)\frac{|x_{i_{j}}-x_{i_{j+1}}|}{|x_{i_{j}}-x_{i_{j}-1}|}+1}$
$\displaystyle\geq\frac{1}{c({\Omega^{(0)}})}.$ (7.7)

For ease of notation let us set $\tilde{x}_{j}=x_{i_{j}}$. Then we shall prove
for $j=J,J-1,\ldots,0$ the following statement, which we call $(\dagger_{j})$:
There is a $\Lambda_{j}({\Omega^{(0)}})$, $V_{j}\in T_{\tilde{x}_{j}}\Omega$,
and $p_{j}\in V_{j}^{\perp}$ so that

$E^{tot}(M,p_{j}+V_{j},\tilde{x}_{j},r)\leq\Lambda_{j}r^{2\alpha}E\quad\forall
r_{j}^{-}\leq r\leq 1/4,$ (7.8)

and

$\theta_{M}(\tilde{x}_{j},r)\leq(7/4)\Theta_{T_{\tilde{x}_{j}}\Omega}\quad\forall
r_{j}^{-}\leq r\leq r_{j}^{+}$ (7.9)

and

$|p_{j}-p|+|\pi_{V_{j}}-\pi_{V}|\leq\Lambda_{j}^{1/2}E^{1/2}(r_{j}^{+})^{\alpha}.$
(7.10)

Observe that (7.8) implies that

$d(x,p_{j}+V_{j})\leq c(r_{j}^{-})^{\alpha}$

and hence when $j=0$, we have $p_{0}+V_{0}=x+V_{0}$. Since $x=\tilde{x}_{0}$,
we have $V_{0}\in T_{x}\Omega$, and (7.10) implies that

$|\pi_{V_{x}}-\pi_{V}|\leq c({\Omega^{(0)}})E^{1/2}.$

Therefore when $j=0$, $(\dagger_{0})$ implies our Theorem.

To prove $(\dagger_{j})$ we induct downwards on $j$. If $j=J$, then let us set
$\tilde{x}_{J+1}=0$, $r_{J+1}^{-}=1/4$, $\Lambda_{J+1}=4^{n+4}$, $p_{J+1}=p$,
$V_{J+1}=V$, and proceed as below. Otherwise, let us assume by hypothesis that
$(\dagger_{j+1})$ holds. We prove $(\dagger_{j})$.

We have

$E^{tot}(\Phi,M,p_{j+1}+V_{j+1},\tilde{x}_{j+1},r)\leq\Lambda_{j+1}r^{2\alpha}E$
(7.11)

for all $r_{j+1}^{-}\leq r\leq 1/4$, and

$\theta_{M}(\tilde{x}_{j+1},r_{j+1}^{-})\leq(7/4)\Theta_{T_{\tilde{x}_{j+1}}\Omega}.$

Since $B_{r_{j}^{+}}(\tilde{x}_{j})\subset B_{r_{j+1}^{-}/2}(\tilde{x}_{j+1})$
and $r_{j}^{+}/r_{j+1}^{-}\geq 1/c({\Omega^{(0)}})$, we can use (7.19) and
apply Lemma 7.2 to deduce

$\theta_{M}(\tilde{x}_{j},r_{j}^{+})\leq(3/2)\Theta_{T_{\tilde{x}_{j}}\Omega},$

and hence by monotonicity (4.1) we get

$\theta_{M}(\tilde{x}_{j},r)\leq(7/4)\Theta_{T_{\tilde{x}_{j}}\Omega}\quad\forall
r\leq r_{j}^{+}.$ (7.12)

(7.11) implies we can find a $q_{j}^{\prime}\in p_{j+1}+V_{j+1}$ so that

$|x-q_{j}^{\prime}|\leq\Lambda_{j+1}^{1/2}E^{1/2}(r_{j+1}^{-})^{1+\alpha}\leq
r_{j+1}^{-},$

for $\delta_{3}(\Lambda_{j+1},{\Omega^{(0)}})$ sufficiently small. In
particular, we have $|\tilde{x}_{j}-q_{j}^{\prime}|\leq cr_{j}^{+}$. As per
Lemma 2.12, choose $V_{j}^{\prime}\in\mathcal{P}_{T_{\tilde{x}_{j}}\Omega}$
such that

$|\pi_{V_{j+1}}-\pi_{V_{j}^{\prime}}|\leq
c({\Omega^{(0)}})E^{1/2}|\tilde{x}_{j}-\tilde{x}_{j+1}|\leq
cE^{1/2}r_{j}^{+}.$ (7.13)

Then, using (7.6) and (7.7), we have

$\displaystyle
E^{\infty}(M,q_{j}^{\prime}+V_{j}^{\prime},\tilde{x}_{j},r_{j}^{+})$ (7.14)
$\displaystyle=(r_{j}^{+})^{-2}\sup_{z\in
B_{r_{j}^{+}}(\tilde{x}_{j})}|\pi_{V_{j}^{\prime}}^{\perp}(z-q_{j}^{\prime})|^{2}$
(7.15)
$\displaystyle=c|\pi_{V_{j}^{\prime}}-\pi_{V_{j+1}}|^{2}+cE^{\infty}(\Phi,M,p_{j+1}+V_{j+1},\tilde{x}_{j+1},r_{j+1}^{-})$
(7.16) $\displaystyle\leq c(1+\Lambda_{j+1})(r_{j}^{+})^{2\alpha}E,$ (7.17)

and similarly

$\displaystyle
E^{W}(M,V_{j}^{\prime},\tilde{x}_{j},r_{j}^{+})+E(\Phi,M,q_{j}^{\prime}+V_{j}^{\prime},\tilde{x}_{j},r_{j}^{+})\leq
c(1+\Lambda_{j+1})(r_{j}^{+})^{2\alpha}E.$ (7.18)

By Lemma 2.12, there is a $z\in{\Omega^{(0)}}$, a linear isomorphism
$\beta:\mathbb{R}^{n+1}\to\mathbb{R}^{n+1}$, and a $C^{2}$ mapping
$\alpha:B_{2}\to\mathbb{R}^{n+1}$ such that
$\Omega_{\tilde{x}_{j},r_{j}^{+}}\in\mathcal{D}_{c\delta_{3}}(T_{x}{\Omega^{(0)}})$
and

$\displaystyle
E^{tot}(\alpha,(\eta_{\tilde{x}_{j},r_{j}^{+}})_{\sharp}M,\eta_{\tilde{x}_{j},r_{j}^{+}}(q_{j}^{\prime}),V_{j}^{\prime},0,1)$
$\displaystyle\leq
2E^{tot}(\Phi,M,q_{j}^{\prime}+V_{j}^{\prime},\tilde{x}_{j},r_{j}^{+})$
$\displaystyle\leq c(1+\Lambda_{j+1})(r_{j}^{+})^{2\alpha}E.$ (7.19)

By (7.12) and (7.19), provided $\delta_{3}({\Omega^{(0)}},\Lambda_{j+1})$ is
sufficiently small, we can apply Corollary 6.2 to deduce: there is a $V_{j}\in
T_{\tilde{x}_{j}}\Omega$, $p_{j}\in V_{j}^{\perp}$, so that

$E^{tot}(\Phi,M,p_{j}+V_{j},\tilde{x}_{j},r)\leq
c(1+\Lambda_{j+1})r^{2\alpha}E\quad\forall r_{j}^{-}\leq r\leq r_{j}^{+},$
(7.20)

(recall that $|x-\tilde{x}_{j}|\leq r_{j}^{-}$) and

$|\pi_{V_{j}}-\pi_{V_{j}^{\prime}}|\leq\left(c(1+\Lambda_{j+1})E\right)^{1/2}(r_{j}^{+})^{\alpha}.$
(7.21)

Combining (7.13), (7.21) we get

$|\pi_{V_{j}}-\pi_{V_{j+1}}|\leq(c(1+\Lambda_{j+1}))^{1/2}E^{1/2}(r_{j}^{+})^{\alpha},$
(7.22)

which with (7.10) implies

$|\pi_{V_{j}}-\pi_{V}|\leq(c(1+\Lambda_{j+1}))^{1/2}E^{1/2}.$ (7.23)

On the other hand, we can estimate

$\displaystyle|p_{j}-p|$
$\displaystyle\leq|\pi_{V_{j}}^{\perp}(x-p_{j})|+|\pi_{V}^{\perp}(x-p)|+|\pi_{V_{j}}-\pi_{V}|$
$\displaystyle\leq
r_{j}^{+}E(M,\Phi,p_{j}+V_{j},\tilde{x}_{j},r_{j})^{1/2}+E(M,\Phi,p+V,0,1)^{1/2}$
$\displaystyle\quad+(c(1+\Lambda_{j+1}))^{1/2}E^{1/2}$
$\displaystyle\leq(c(1+\Lambda_{j+1}))^{1/2}E^{1/2}.$ (7.24)

Finally, we must show decay (7.20) for $r\geq r_{j}^{+}$. (7.20) implies that
we can find a $q_{j}\in p_{j}+V_{j}$ such that

$|x-q_{j}|\leq(c(1+\Lambda_{j+1}))^{1/2}E^{1/2}(r_{j}^{+})^{1+\alpha}\leq
r_{j}^{+}$

and hence

$|q_{j}-q_{j}^{\prime}|\leq(c(1+\Lambda_{j+1}))^{1/2}E^{1/2}(r_{j}^{+})^{1+\alpha},\quad\text{
and }\quad|\tilde{x}_{j}-q_{j}|\leq cr_{j}^{+}.$ (7.25)

Therefore, using (7.22), (7.25), (7.6), (7.7), we have for $r_{j}^{+}\leq
r\leq(1/4)(r_{j}^{+}/r_{j+1}^{-})$:

$\displaystyle E^{\infty}(\Phi,M,p_{j}+V_{j}\equiv
q_{j}+V_{j},\tilde{x}_{j},r)$ $\displaystyle=r^{-2}\sup_{z\in
B_{r}(\tilde{x}_{j})\cap\mathrm{spt}M}|\pi_{V_{j}}(z-q_{j})|^{2}$
$\displaystyle\leq
c|\pi_{V_{j}}-\pi_{V_{j+1}}|^{2}+r^{-2}|q_{j}-q_{j}^{\prime}|+cE^{\infty}(\Phi,M,q_{j}^{\prime}+V_{j+1},\tilde{x}_{j+1},rr^{-}_{j+1}/r_{j}^{+})$
$\displaystyle\leq c(1+\Lambda_{j+1})Er^{2\alpha}.$

On the other hand, when $(1/4)r_{j}^{+}/r_{j+1}^{-}\leq r\leq 1/4$, then we
have $r\geq 1/c({\Omega^{(0)}})$, and hence we can estimate using (7.23),
(7.24):

$\displaystyle E^{\infty}(\Phi,M,p_{j}+V_{j},\tilde{x}_{j},r)$
$\displaystyle\leq
c|\pi_{V_{j}}-\pi_{V}|^{2}+c|p_{j}-p|^{2}+E^{\infty}(\Phi,M,p+V,0,1)$
$\displaystyle\leq c(1+\Lambda_{j+1})E$ $\displaystyle\leq
c(1+\Lambda_{j+1})Er^{2\alpha}.$

Bounds on $E^{W}$ and $E$ follow by similar computations.

Provided we take $\Lambda_{j}\geq c(1+\Lambda_{j+1})$ sufficiently big,
depending only on $c({\Omega^{(0)}})$ and $\Lambda_{j+1}$, this proves
$(\dagger_{j})$. ∎

###### Proof of Theorem 3.1.

Take $\delta({\Omega^{(0)}})$ sufficiently small so that we can apply Theorem
7.1. Given $x\in\mathrm{spt}M\cap B_{1/16}$, let
$V_{x}\in\mathcal{P}_{T_{x}\Omega}$ be as in Theorem 7.1.

Given $x,y\in\mathrm{spt}M\cap B_{1/16}$, using (4.30) and (7.3) we can
estimate

$\displaystyle\frac{1}{c({\Omega^{(0)}})}|\pi_{V_{x}}-\pi_{V_{y}}|^{2}$
$\displaystyle\leq|x-y|^{-n}\mu_{M}(B_{|x-y|}(x))|\pi_{V_{x}}-\pi_{V_{y}}|^{2}$
$\displaystyle\leq
2|x-y|^{-n}\int_{B_{|x-y|}(x)}|\pi_{V_{x}}-\pi_{T_{z}M}|^{2}+|\pi_{V_{y}}-\pi_{T_{z}M}|^{2}d\mu_{M}$
$\displaystyle\leq
2E^{tot}(\Phi,M,x+V_{x},x,|x-y|)+2E^{tot}(\Phi,M,y+V_{y},y,2|x-y|)$
$\displaystyle\leq c({\Omega^{(0)}})|x-y|^{2\alpha}E,$

and hence

$|\pi_{V_{x}}-\pi_{V_{y}}|\leq c({\Omega^{(0)}})E^{1/2}|x-y|^{\alpha}.$ (7.26)

This effectively shows that $\mathrm{spt}M$ lies inside some $C^{1,\alpha}$
graph. We must show that $\mathrm{spt}M$ fills out this entire graph, and the
same holds true for $(\Phi^{-1})_{\sharp}M$. We first prove two auxillary
claims.

Claim 1: For all $x\in\mathrm{int}\Omega\cap\mathrm{spt}M\cap B_{1/16}$, we
have

$\frac{1}{r}d_{H}(\mathrm{spt}M\cap B_{r}(x),(x+V_{x})\cap B_{r}(x))\to
0\quad\text{ as }r\to 0.$

We prove this by contradiction. Otherwise, suppose there is a sequence
$r_{i}\to 0$ and $\epsilon>0$ so that

$\displaystyle\frac{1}{2r_{i}}d_{H}(\mathrm{spt}M\cap
B_{r_{i}}(x),(x+V_{x})\cap B_{r_{i}}(x))$ $\displaystyle\equiv
d_{H}(\mathrm{spt}M_{i}\cap B_{1/2},V_{x}\cap B_{1/2})$
$\displaystyle\geq\epsilon\quad\forall i.,$ (7.27)

where $M_{i}=(\eta_{x,r_{i}})_{\sharp}M$. Since
$E^{\infty}(M_{i},V_{x},0,1)\to 0$ as $r\to 0$, we can argue as in Proposition
6.4, using Theorem 4.3 and Corollary 4.8, to deduce that (after passing to a
subsequence)

$M_{i}\to[V_{x}]$

as varifolds in $B_{1}$. The lower bound (4.30) then implies that
$\mathrm{spt}M_{i}\cap B_{1/2}\to V_{x}\cap B_{1/2}$ in the Hausdorff
distance, which is a contradiction for large $i$. This proves Claim 1.

Claim 2: We have $\mathrm{spt}M\cap
B_{1/512}\cap\mathrm{int}\Omega\neq\emptyset$.

On the one hand, by (4.30) we have $\lim_{r\to 0}r^{-n}\mu_{M}(B_{r}(x))>0$
for every $x\in\mathrm{spt}M$. On the other hand, for every
$x\in\mathrm{spt}M\cap B_{1/16}\cap\partial\Omega$ the decay bound (7.3) and
non-concentration estimate (4.34) imply

$\limsup_{r\to 0}r^{-n}\mu_{M}(B_{r}(x)\cap\partial\Omega)=0.$

Since $\mathrm{spt}M\cap B_{1/512}\neq\emptyset$, this proves Claim 2.

Let $M_{\Phi}=(\Phi^{-1})_{\sharp}M$. Since $1/c\leq\theta_{M}(x)\leq c$ for
every $x\in\mathrm{spt}M\cap B_{\theta}$, $c=c(\theta,{\Omega^{(0)}})$, we can
write

$\mu_{M}=\mathcal{H}^{n}\llcorner\mathrm{spt}M\llcorner\theta_{M},$

where $\mathrm{spt}M$ is countably-$n$-rectifiable. By the area formula we can
therefore write

$\mu_{M_{\Phi}}=\mathcal{H}^{n}\llcorner\Phi^{-1}(\mathrm{spt}M)\llcorner(\theta_{M}\circ\Phi).$

This implies (for $\delta(n)$ sufficiently small) that
$\Phi(\mathrm{spt}M)\cap B_{1/32}=\mathrm{spt}M_{\Phi}\cap B_{1/32}$,

$\Phi(\mathrm{spt}M_{\Phi}\cap B_{1/32})\subset\mathrm{spt}M\cap B_{1/16},$

and $T_{\Phi^{-1}(x)}M_{\Phi}=D\Phi^{-1}|_{x}T_{x}M$ for $\mu_{M}$-a.e. $x\in
B_{1/16}$.

We aim to show decay estimates like (7.3), (7.26) for
$\Phi^{-1}(\mathrm{spt}M)\equiv\mathrm{spt}M_{\Phi}$. This is essentially a
direct consequence of the fact that $\Phi$ is a $C^{2}$ diffeomorphism. Define
$q=\Phi^{-1}(p)$, $W=D\Phi^{-1}|_{p}V$ for $p$, $V$ as in our hypotheses. For
$z\in\mathrm{spt}M_{\Phi}\cap B_{1/32}$, define
$W_{z}=D\Phi^{-1}|_{\Phi(z)}V_{\Phi(z)}$.

Let us recall that $\Phi$ is a bi-Lipschitz equivalence: for every $x,y\in
B_{1}$ we have

$|\Phi(x)-\Phi(y)-(x-y)|\leq c(n)\delta|x-y|,$

and we have the bound

$|D\Phi^{-1}|_{x}-D\Phi^{-1}|_{y}|\leq c(n)|D^{2}\Phi|_{C^{0}(B_{1})}|x-y|.$

Claim 3: Take $a\in B_{1/16}$, $V$ an $n$-plane, $y\in\mathrm{spt}M\cap
B_{r}(a)$ for $r<1/4$, and $x\in(a+V)\cap B_{r}(a)$. Then we can find a
$\tilde{x}\in(\Phi^{-1}(a)+D\Phi^{-1}|_{a}(V))\cap
B_{(1+c(n)\delta)r}(\Phi^{-1}(a))$ such that

$|\Phi^{-1}(y)-\tilde{x}|\leq 2|y-x|+c(n)|D^{2}\Phi|_{C^{0}(B_{2})}r^{2}.$

To prove this, let $\tilde{x}=\Phi^{-1}(a)+D\Phi^{-1}|_{a}(x-a)$. Then

$|\Phi^{-1}(x)-\tilde{x}|=|\Phi^{-1}(x)-\Phi^{-1}(a)-D\Phi^{-1}_{a}(x-a)|\leq
c(n)|D^{2}\Phi|_{C^{0}(B_{2})}|x-a|^{2},$

and

$|\Phi^{-1}(y)-\Phi^{-1}(x)|\leq(1+c\delta)|y-x|.$

This proves Claim 3.

Claim 3 and our definition of $W_{\tilde{a}}$ implies that for every
$\tilde{a}=\Phi(a)\in\mathrm{spt}M_{\Phi}\cap B_{1/32}$ and $r<1/8$ we have

$\displaystyle\sup_{z\in\mathrm{spt}M_{\Phi}\cap
B_{r}(\tilde{a})}d(z,\tilde{a}+W_{\tilde{a}})$ $\displaystyle\leq
2\sup_{y\in\mathrm{spt}M\cap B_{2r}(a)}d(y,a+V_{a})+c(n)|D^{2}\Phi|r^{2}$
$\displaystyle\leq c(n)E^{\infty}(\Phi,M,a+V_{a},a,2r)^{1/2}r$
$\displaystyle\leq c({\Omega^{(0)}})E^{1/2}r^{1+\alpha}$ (7.28)

and by the same reasoning

$\sup_{z\in\mathrm{spt}M_{\Phi}\cap B_{1/32}}d(z,q+W)\leq
c({\Omega^{(0)}})E^{1/2}.$ (7.29)

Similarly, combining Claim 1 with Claim 3 we have that

$\lim_{r\to 0}\frac{1}{r}d_{H}(\mathrm{spt}M_{\Phi}\cap
B_{r}(\tilde{a}),(\tilde{a}+W_{\tilde{a}})\cap B_{r}(\tilde{a}))=0$ (7.30)

for every $\tilde{a}\in\mathrm{spt}M_{\Phi}\cap
B_{1/32}\cap\mathrm{int}\Omega$.

Like in the proof of Lemma 2.12, we have the bounds

$|\pi_{A(V)}-\pi_{B(V)}|\leq c(n)|A-B|,\quad|\pi_{A(V)}-\pi_{A(W)}|\leq
c(n)|\pi_{V}-\pi_{W}|$

for linear maps $A,B$ satisfying $|A-Id|+|B-Id|\leq\epsilon(n)$. Therefore for
$\delta(n)$ sufficiently small, we can estimate

$\displaystyle|\pi_{W_{\tilde{a}}}-\pi_{W_{\tilde{b}}}|$ $\displaystyle\leq
c(n)|D\Phi^{-1}|_{a}-D\Phi^{-1}|_{b}|+c(n)|\pi_{V_{a}}-\pi_{V_{b}}|$
$\displaystyle\leq
c(n)|D^{2}\Phi|_{C^{0}(B_{1})}|a-b|+c({\Omega^{(0)}})E^{1/2}|a-b|^{\alpha}$
$\displaystyle\leq c({\Omega^{(0)}})E^{1/2}|\tilde{a}-\tilde{b}|^{\alpha},$
(7.31)

where $\tilde{a}=\Phi(a)$, $\tilde{b}=\Phi(b)$ both lie in
$\mathrm{spt}M_{\Phi}\cap B_{1/32}$. The same proof gives us the bound

$|\pi_{W_{\tilde{a}}}-\pi_{W}|\leq c({\Omega^{(0)}})E^{1/2}.$ (7.32)

Combining (7.28), (7.31), (7.32) we get

$\displaystyle|\pi_{W}^{\perp}(y-z)|\leq
c({\Omega^{(0)}})E^{1/2}|x-y|,\quad|\pi_{W_{z}}^{\perp}(y-z)|\leq
c({\Omega^{(0)}})E^{1/2}|y-z|^{1+\alpha},$ (7.33)
$\displaystyle\quad|\pi_{W_{y}}-\pi_{W_{z}}|\leq
c({\Omega^{(0)}})E^{1/2}|y-z|^{\alpha}$

for all $y,z\in\mathrm{spt}M_{\Phi}\cap B_{1/32}$.

Let $U_{0}=\Omega^{\prime}\cap(q+W)\cap\overline{B_{1/128}(q)}$,
$\mathrm{int}U_{0}=\mathrm{int}\Omega^{\prime}\cap(q+W)\cap B_{1/128}(q)$, and
$U=(U_{0}\times W^{\perp})\cap B_{1/64}(q)$. For $\delta({\Omega^{(0)}})$
sufficiently small we have $|q|\leq 1/300$, and hence

$B_{1/256}\subset U\subset B_{1/32}.$ (7.34)

Further, by (7.29) we can assume that

$\mathrm{spt}M_{\Phi}\cap B_{1/32}\subset B_{1/300}(q+W),$ (7.35)

and therefore $\mathrm{spt}M_{\Phi}\cap U$ is a closed subset of
$\mathbb{R}^{n+1}$.

By (7.33), (7.35) the projection mapping $F(x)=\pi_{W}(x)+\pi_{W}^{\perp}(q)$
induces a $(1+c\delta)$-bi-Lipschitz equivalence between
$\mathrm{spt}M_{\Phi}\cap U$ and some closed subset $\tilde{U}_{0}\subset
U_{0}$, and in particular by (7.30) $\tilde{U}_{0}$ satisfies the property
that

$\lim_{r\to 0}d_{H}(B_{r}(y)\cap\tilde{U}_{0},B_{r}(y)\cap
U_{0})=0\quad\forall y\in\tilde{U}_{0}\cap\mathrm{int}U_{0}.$

It follows by an elementary argument that we must have either
$\tilde{U}_{0}\cap\mathrm{int}U_{0}=\mathrm{int}U_{0}$ or
$\tilde{U}_{0}\cap\mathrm{int}U_{0}=\emptyset$. By Claim 2
$\mathrm{spt}M_{\Phi}\cap\mathrm{int}\Omega^{\prime}\cap
B_{1/256}\neq\emptyset$, so $\tilde{U}_{0}\cap\mathrm{int}U_{0}\neq\emptyset$,
and hence $\tilde{U}_{0}=U_{0}$.

This proves $F$ is a $(1+c\delta)$-bi-Lipschitz equivalence between
$\mathrm{spt}M\cap U$ and $U_{0}$. We can therefore find a Lipschitz function
$u:U_{0}\to\mathbb{R}$ so that $\mathrm{spt}M_{\Phi}\cap U=\mathrm{graph}(u)$.
It follows easily from (7.33), (7.29) that $u$ is $C^{1,\alpha}$ and satisfies
the bound $|u|_{C^{1,\alpha}(U_{0})}\leq c({\Omega^{(0)}})E^{1/2}$. ∎

##  8\. Higher codimension and ambient manifolds

The only part of our regularity theorem that requires codimension-one is in
obtaining the estimates (4.13), (4.19). If $l=1$, or one can otherwise verify
these estimates, then we get corresponding regularity in higher codimension.