such that $g(\eta_{V_{I}},\nu_{g})=\cos\beta(x)$ for
$\mathcal{H}^{n-1}_{g}$-a.e. $x\in BS$. Note that a direct consequence of
(3.121) and (3.10) is that, in $B_{\gamma}$:

$d\sigma_{V,g}=\cos\beta(x)d\mathcal{H}^{n-1}_{g}\llcorner
BS+d\mathcal{H}^{n-1}_{g}\llcorner(\mathrm{spt}\sigma_{V,g}\setminus BS).$
(3.122)

######  Remark 3.17.

We think it likely that Theorem 3.16 holds with $\mathrm{spt}\sigma_{V_{B}}$
in place of $BS$. The issue is ruling out the possibility that
$(\overline{\partial^{*}S}\setminus\partial^{*}S)\cap\\{\beta\neq\pi/2\\}$ has
positive $\mathcal{H}^{n-1}_{g}$-measure. See Remark 1.20 for more discussion.

###### Proof of Theorem 3.16.

Ensuring $\gamma(n,\alpha)$, $\delta(n,\alpha)$ are sufficiently small, by
monotonicity (2.62) and Remark 2.6 we can assume

$\Theta_{V}(x,r)+(\cos\beta(x))_{-}\leq 1-\alpha/4\quad\forall x\in
B_{\gamma}\cap\mathbb{R}^{n},\quad\forall r\in(0,1-|x|).$ (3.123)

Ensure also that $\gamma$ is smaller than the $\gamma(n,\alpha/2)$ from
Theorem 3.1. Let us recall that for some $c=c(n,\alpha)$ we have from Theorem
3.1.

$c^{-1}\mathcal{H}^{n-1}_{g}\leq\sigma_{V,g}\leq c\mathcal{H}^{n-1}_{g},\quad
c^{-1}\sigma_{V_{I},g}\leq\sigma_{V,g}\leq c\sigma_{V_{I},g}\quad\text{ on
}B_{\gamma}.$ (3.124)

Choose an $x\in\mathrm{spt}V_{I}\cap\mathbb{R}^{n}\cap
B_{\gamma}\equiv\mathrm{spt}\sigma_{V}\cap B_{\gamma}$, and suppose
$g(x)=g_{eucl}$. From (3.123), Lemma 3.12, and monotonicity Lemma 2.9, for any
$r_{i}\to 0$, up to a subsequence the dilations
$V_{i}:=(\eta_{x,r_{i}})_{\sharp}V\to V^{\prime}$ (where each $V_{i}$ lives in
the metric $g_{i}=g\circ\eta_{x,r_{i}}^{-1}$) for $V^{\prime}$ a conical,
$(\beta^{\prime}\equiv\beta(x),S^{\prime})$-capillary varifold in
$\mathbb{R}^{n+1}$ satisfying

$\Theta_{V^{\prime}}(0,1)=\Theta_{V,g}(x)\leq
1-\alpha/4-(\cos\beta^{\prime})_{-}.$ (3.125)

Moreover, we have $V_{I,i}\to V_{I}^{\prime}$, $V_{B,i}\to V_{B}^{\prime}$,
and $\sigma_{V_{i},g_{i}}\to\sigma_{V^{\prime}}$.

Since $\Theta_{V,g}$ is upper-semi-continuous on $B_{1}\cap\mathbb{R}^{n}$,
for $\sigma_{V,g}$-a.e. $x$ we can suppose $x$ is a Lebesgue point of
$\Theta_{V,g}|_{\mathrm{spt}\sigma_{V,g}\cap B_{1}}$ w.r.t. $\sigma_{V,g}$.
Fix such an $x$. Then we claim we must have
$\Theta_{V^{\prime}}(y)=\Theta_{V^{\prime}}(0)$ for all
$y\in\mathrm{spt}\sigma_{V^{\prime}}$. Otherwise, if
$y\in\mathrm{spt}\sigma_{V^{\prime}}$ satisfies
$\Theta_{V^{\prime}}(y)<\Theta_{V^{\prime}}(0)$, then for some $\varepsilon>0$
we have $\Theta_{V_{i},g_{i}}(z)<\Theta_{V^{\prime}}(0)-\varepsilon$ for all
$z\in B_{\varepsilon}(y)$ and all $i\gg 1$. From Theorem 3.1 with $R=|y|+1$ we
have

$\varepsilon^{n-1}/c(n,\alpha)\leq\sigma_{V_{i},g_{i}}(B_{\varepsilon}(y))\leq\sigma_{V_{i},g_{i}}(\overline{B_{R}})\leq
c(n,\alpha)R^{n-1}$

for all $i\gg 1$. We then deduce the contradiction

$\displaystyle\fint_{\overline{B_{Rr_{i}}(x)}}|\Theta_{V,g}-\Theta_{V,g}(x)|d\sigma_{V,g}$
$\displaystyle=\fint_{\overline{B_{R}}}|\Theta_{V_{i},g_{i}}-\Theta_{V_{i},g_{i}}(0)|d\sigma_{V_{i},g_{i}}$
(3.126)
$\displaystyle\geq\frac{\sigma_{V_{i},g_{i}}(\overline{B_{\varepsilon}(y))}}{\sigma_{V_{i},g_{i}}(\overline{B_{R}})}$
(3.127) $\displaystyle\geq 1/c(n,\alpha,\varepsilon,y).$ (3.128)

This proves our claim.

Since $\mathrm{spt}\sigma_{V^{\prime}}\neq\emptyset$ and
$\mathcal{H}^{n-1}(\mathrm{spt}\sigma_{V^{\prime}}\cap B_{1})>0$, we deduce by
monotonicity that $V^{\prime}$ has at least $(n-1)$-dimensions of
translational symmetry. Using (3.125), that
$V^{\prime}_{B}=\cos\beta^{\prime}[S^{\prime}]\equiv\cos\beta(x)[S^{\prime}]$,
and that $V^{\prime}_{I}$ is integral, we deduce that (up to rotation)

$\displaystyle V^{\prime}=V^{(\beta^{\prime})}\quad\text{ or }\quad
V^{\prime}=V^{(\pi/2)}\quad\text{ or }\quad
V^{\prime}=V^{(\pi/2)}+\cos\beta^{\prime}[\mathbb{R}^{n}].$ (3.129)

Note that the nature of $V^{\prime}$ (though not, a priori, the rotation) is
uniquely determined by $\Theta_{V,g}(x)$ being one of
$1/2,(1+\cos\beta(x))/2,1/2+\cos\beta(x)$.

Observe also that we have
$\sigma_{V_{I}^{\prime}}=\mathcal{H}^{n-1}\llcorner\mathbb{R}^{n-1}$ and
either $\sigma_{V^{\prime}_{B}}=0$ or
$\sigma_{V_{B}^{\prime}}=\cos\beta^{\prime}\mathcal{H}^{n-1}\llcorner\mathbb{R}^{n-1}$.
So at $\sigma_{V,g}$-a.e. $x$, every tangent measure of $\sigma_{V,g}$ is a
multiple and rotation of $\mathcal{H}^{n-1}\llcorner\mathbb{R}^{n-1}$. Along
with the Ahlfors-regularity (3.6), it follows from the Marstrand–Mattila
rectifiability criterion ([4, Theorem 5.1]) that $\sigma_{V,g}$ is
$(n-1)$-rectifiable, and hence (by (3.124)) $\mathrm{spt}\sigma_{V,g}$ is
$(n-1)$-rectifiable.

By standard density estimates ([23, Chapter 1]), for
$\mathcal{H}^{n-1}_{g}$-a.e. $x\in
B_{\gamma}\cap\\{\beta\neq\pi/2\\}\setminus\partial^{*}S$ we have

$\mu_{V^{\prime}_{B}}(B_{1})\leq\limsup_{r\to
0}\frac{\mathcal{H}^{n-1}_{g}(\partial^{*}S\cap B_{r}(x))}{r^{n-1}}=0,$

which implies $V^{\prime}=V^{(\pi/2)}$ or
$V^{\prime}=V^{(\pi/2)}+\cos\beta^{\prime}[\mathbb{R}^{n}]$. On the other
hand, from De Giorgi’s structure theorem, for $\mathcal{H}^{n-1}_{g}$-a.e.
$x\in B_{\gamma}\cap\\{\beta\neq\pi/2\\}\cap\partial^{*}S$ we have

$\mu_{V^{\prime}_{B}}(B_{1})=\lim_{r\to
0}\frac{\cos\beta(x)\mathcal{H}^{n}_{g}(S\cap
B_{r}(x))}{r^{n}}=\cos\beta(x)\omega_{n}/2,$

which implies $V^{\prime}=V^{(\beta(x))}$. Of course we have
$V^{\prime}=V^{(\pi/2)}$ or
$V^{\prime}=V^{(\pi/2)}+\cos\beta(x)[\mathbb{R}^{n}]$ for $\sigma_{V,g}$-a.e.
$x\in B_{\gamma}\cap\\{\beta=\pi/2\\}$.

For $\sigma_{V_{I},g}$-a.e. $x\in B_{1}$ (and hence by (3.124) for
$\sigma_{V,g}$-a.e. $x\in B_{1}$), we have $\eta_{V_{I},g}(x)=\lim_{r\to
0}\fint_{\overline{B_{r}(x)}}\eta_{V_{I},g}d\sigma_{V_{I},g}$. Fix an $x$
satisfying both this and being a Lebesgue point for
$\Theta_{V,g}|_{\mathrm{spt}\sigma_{V,g}}$ w.r.t. $\sigma_{V,g}$. Taking the
tangent cone as above, we can assume the rescaled measures
$\sigma_{V_{I,i},g_{i}}\to\lambda$ as Radon measures for some
$\lambda\geq\sigma_{V_{I}^{\prime}}$. We then compute for a.e. $R$:

$\displaystyle\lambda(\overline{B_{R}})$
$\displaystyle=\lim_{i\to\infty}\sigma_{V_{I,i},g_{i}}(\overline{B_{R}})$
(3.130)
$\displaystyle=\lim_{i\to\infty}\int_{\overline{B_{R}}}\eta_{V_{I},g}(x)\cdot\eta_{V_{I,i},g_{i}}(y)d\sigma_{V_{I,i},g_{i}}(y)$
(3.131)
$\displaystyle=\int_{\overline{B_{R}}}\eta_{V_{I},g}(x)\cdot\eta_{V_{I}^{\prime}}(y)d\sigma_{V_{I}^{\prime}}(y)$
(3.132) $\displaystyle\leq\sigma_{V_{I}^{\prime}}(\overline{B_{R}}).$ (3.133)

Therefore $\sigma_{V_{I,i},g_{i}}\to\sigma_{V_{I}^{\prime}}$ as Radon
measures, and $\eta_{V_{I}^{\prime}}\equiv\eta_{V_{I},g}(x)$. In particular,
we have $\lim_{r\to 0}r^{-n+1}\sigma_{V_{I}}(B_{r}(x))=\omega_{n-1}$ for
$\sigma_{V}$-a.e. $x\in B_{\gamma}$.

From our characterization of tangent cone we deduce that
$\eta_{V_{I},g}(x)=\nu_{g}$ for $\sigma_{V,g}$-a.e. $x\in
B_{\gamma}\setminus(\\{\beta\neq\pi/2\\}\cap\partial^{*}S)$, and
$g(\eta_{V_{I},g}(x),\nu_{g})=\cos\beta(x)$ for $\sigma_{V,g}$-a.e. $x\in
B_{\gamma}\cap\\{\beta\neq\pi/2\\}\cap\partial^{*}S$. Finally, using that
$\mathrm{spt}\sigma_{V,g}$ is rectifiable we deduce that

$\lim_{r\to
0}\frac{\sigma_{V_{I},g}(\overline{B_{r}(x)})}{\mathcal{H}^{n-1}_{g}(\mathrm{spt}\sigma_{V}\cap\overline{B_{r}(x)})}=\lim_{r\to
0}\frac{\omega_{n-1}r^{n-1}}{\mathcal{H}^{n-1}_{g}(\mathrm{spt}\sigma_{V,g}\cap\overline{B_{r}(x)})}\frac{\sigma_{V_{I},g}(\overline{B_{r}(x)})}{\omega_{n-1}r^{n-1}}=1$

for $\sigma_{V,g}$-a.e. $x\in B_{\gamma}$. The characterization (3.121) now
follows by the Radon-Nikodyn theorem. ∎

##  4\. Boundary maximum principle

Our goal of this section is to establish the following boundary maximum
principle (Theorem 4.1). Our strategy is to first prove the maximum principle
at the level of the tangent cone (Lemma 4.5), by showing that any blow-up
cannot be contained in a wedge of too small or large an angle. To ensure the
tangent cone has genuine capillary boundary data (as opposed to secretly being
free-boundary), we employ an argument (Lemma 4.4) which shows that closeness
to the model cone $V^{(\theta)}$ propagates to all scales on the boundary
_provided_ one allows $V^{(\theta)}$ to rotate with the scale. The argument is
very soft and only uses monotonicity and a fairly basic classification of
cones (Lemma 4.2).

In Sections 4, 5, 6 we will always be working with capillary angles
$\in(0,\pi/2)$, and in particular our capillary varifolds will always have
positive density. Otherwise, if $\beta>\pi/2$ locally and
$V=V_{I}+\cos\beta[S]_{g}$, then we work with the swapped varifold
$\tilde{V}=V_{I}+\cos(\pi-\beta)[\mathbb{R}^{n}\setminus S]_{g}$ instead.

Before stating Theorem 4.1, we recall the notations $G_{\theta}(u)$ and
$\nu_{\theta}(u,g)$ introduced in Subsection 2.1 and we introduce the
following (straightforward) terminology: we say that $G_{\theta}(u)$ lies
above (resp. below) $\mathrm{spt}\mu_{V_{I}}$ in some set $E$ if
$\mathrm{spt}\mu_{V_{I}}\cap E\subset\\{x\colon x_{n+1}\leq-
x_{1}\cot\theta+u(x^{\prime})\mbox{ (resp. $\geq$)}\\}$ and that it touches
$\mathrm{spt}\mu_{V_{I}}$ at $z$ if $z\in\mathrm{spt}\mu_{V_{I}}\cap
G_{\theta}(u)$.

######  Theorem 4.1 (Boundary maximum principle).

Given $\theta\in(0,\pi/2)$, there is a $\delta(n,\theta)$ so that the
following holds. Let $g$ be a $C^{1}$ metric on $B_{1}$, $\beta$ be a $C^{1}$
function on $B_{1}$, and $V$ a $(\beta,S)$-capillary varifold in $(B_{1},g)$
satisfying

$\displaystyle\mathcal{D}(V,V^{(\theta)})\leq\delta,$ (4.1)
$\displaystyle\max\\{||H^{tan}_{V,g}||_{L^{\infty}(B_{1})},|g-g_{eucl}|_{C^{1}(B_{1})},|\beta-\theta|_{C^{1}(B_{1})}\\}\leq\delta.$
(4.2)

Let $u:B_{1/2}^{\theta}\to\mathbb{R}$ be a $C^{1}$ function, and
$z\in\mathrm{spt}\mu_{V_{I}}\cap\\{x_{1}=0\\}\cap B_{3/4}$. Denote by
$G_{\theta}(u)$ the slanted graph of $u$ and $\nu_{g}$ the $g$-outward unit
normal vector field along $\\{x_{1}=0\\}$.

  1. (1)

If $G_{\theta}(u)$ lies above $\mathrm{spt}\mu_{V_{I}}$ in
$C_{1/2}^{\theta}\cap B_{3/4}$ and touches $\mathrm{spt}\mu_{V_{I}}$ at $z$,
then

$g(\nu_{\theta}(u,g)|_{z},\nu_{g}|_{z})\geq\cos\beta(z).$ (4.3)

  2. (2)

If $G_{\theta}(u)$ lies below $\mathrm{spt}\mu_{V_{I}}$ in
$C_{1/2}^{\theta}\cap B_{3/4}$ and touches $\mathrm{spt}\mu_{V_{I}}$ at $z$,
then

$g(\nu_{\theta}(u,g)|_{z},\nu_{g}|_{z})\leq\cos\beta(z).$ (4.4)

######  Lemma 4.2 (Classification of tangent cones).

Let $\theta\in(0,\pi/2)$ and let $V$ be a conical, stationary
$(\theta,S)$-capillary varifold in $\mathbb{R}^{n+1}$ satisfying

$\displaystyle\Theta_{V}(0)\leq(1+\cos\theta)/2,$ (4.5) $\displaystyle
0<\mu_{V}(B_{1}\cap\mathbb{R}^{n})<\cos\theta\omega_{n},$ (4.6)

then $V=q(V^{(\theta)})$ for some $q\in O(\mathbb{R}^{n})$.

######  Remark 4.3.

This Lemma fails without the assumption (4.6). For a counterexample, take
$S=\emptyset$ and $V$ to be half the Simons’ cone. Then (4.5) holds for some
$\theta\in(0,\pi/2)$ and $\mu_{V}(B_{1}\cap\mathbb{R}^{n})=0$.

###### Proof of Lemma 4.2.

We prove this by induction on $n$. If $n=1$ then $V$ consists of finitely-many
rays extending from the origin and the conclusion follows directly by our
assumptions. Suppose $n>1$ and that the Theorem holds for $n-1$.

By Theorem 3.1 $S$ is a set of locally-finite perimeter, and
$\partial^{*}S\subset\mathrm{spt}\sigma_{V}$, and (by (4.6))
$\partial^{*}S\neq\emptyset$. Therefore for every $x\in\partial^{*}S$, after a
rotation as necessary the rescaled sets $(S-x)/r$ converge in
$L^{1}_{loc}(\mathbb{R}^{n})$ to $\\{x_{1}=0,x_{n+1}<0\\}$. Then by Lemma 3.12
and our upper density assumption (4.5) any tangent cone $V^{\prime}$ to $V$ at
$x$ is a $(\theta,S^{\prime})$-capillary varifold which satisfies
$V^{\prime}\llcorner\\{x_{1}=0\\}=\cos\theta\mathcal{H}^{n}\llcorner\\{x_{1}=0,x_{n+1}<0\\}$.
Moreover, if $x\neq 0$, then $V^{\prime}$ must have a line of translational-
invariance, and in particular we can write $V^{\prime}=[\mathbb{R}]\times
V^{\prime\prime}$ for $V^{\prime\prime}$ satisfying the same hypotheses (4.5),
(4.6) except with $n-1$ in place of $n$. By inductive hypothesis
$\Theta_{V^{\prime\prime}}(x)=(1+\cos\theta)/2$.

By upper semicontinuity, we deduce that $\Theta_{V}\geq(1+\cos\theta)/2$ on
$\overline{\partial^{*}S}$, but since $\mathcal{H}^{n-1}(\partial^{*}S)>0$ we
have by standard splitting that $V$ has $(n-1)$-dimensions of translational-
invariance, and hence $V$ has the required form. ∎

######  Lemma 4.4 (Propagation of closeness to all scales).

Given $\theta\in(0,\pi/2)$, $\varepsilon>0$, and $\sigma\in(0,1)$, there is
$\delta(n,\theta,\varepsilon,\sigma)>0$ so that the following holds. Let $g$
be a $C^{1}$ metric on $B_{1}$, $\beta$ a $C^{1}$ function on $B_{1}$, and let
$V$ be a $(\beta,S)$-capillary varifold in $(B_{1},g)$ supported in
$\\{x_{1}\leq 0\\}$ satisfying

$\displaystyle\mathcal{D}(V,V^{(\theta)})\leq\delta,$ (4.7)
$\displaystyle\max\\{||H^{tan}_{V,g}||_{L^{\infty}(B_{1})},|g-g_{eucl}|_{C^{1}(B_{1})},|\beta-\theta|_{C^{1}(B_{1})}|\\}\leq\delta.$
(4.8)

Then for all $x\in B_{\sigma}\cap\mathrm{spt}\mu_{V_{I}}\cap\\{x_{1}=0\\}$ and
$r\in(0,1-\sigma)$, we have

$\displaystyle\inf_{q\in
O(\\{x_{1}=0\\})}\mathcal{D}((V-x)/r,q(V^{(\theta)}))\leq\varepsilon,$ (4.9)

and the same inequality (4.9) holds with $V^{(\beta(x))}$ in place of
$V^{(\theta)}$.

###### Proof.

We prove (4.9) for $V^{(\theta)}$. The case for $V^{(\beta(x))}$ follows
immediately from (4.8) and the observation that
$\mathcal{D}(V^{(\beta(x))},V^{(\theta)})\leq\varepsilon^{\prime}$ whenever
$\delta(n,\varepsilon^{\prime},\theta)$ is sufficiently small. Also we note
there is no loss in assuming $\varepsilon(n,\theta)$ is as small as we like.

Suppose the statement fails. Then there exist sequences $\delta_{i}\to 0$,
$r_{i}\in(0,1-\sigma)$, varifolds $V_{i}$ in $(B_{1},g_{i})$ and $y_{i}\in
B_{\sigma}\cap\mathrm{spt}\mu_{V_{i,I}}\cap\\{x_{1}=0\\}$, such that the
assumptions (4.7), (4.8) hold with $V_{i},g_{i},\delta_{i}$ in place of
$V,g,\delta$, while

$\mathcal{D}((V_{i}-y_{i})/r_{i},q(V^{(\theta)}))>\varepsilon$

for all $q\in O(n)$. Note that we must have that $r_{i}\to 0$, since (4.7)
implies that $V\to V^{(\theta)}$ in any fixed ball.

Note also that, by Theorem 3.1 applied in any ball $B_{(1-\sigma)/2}(x),x\in
B_{\sigma}\cap\mathbb{R}^{n}$, it holds

$\mathrm{spt}\sigma_{V_{i},g_{i}}\cap
B_{\sigma}=\mathrm{spt}\mu_{V_{i,I}}\cap\mathbb{R}^{n}\cap B_{\sigma}$

. Moreover, (4.7), (4.8) and Lemma 3.12 imply that
$\Theta_{V_{i}}(x,(1-\sigma)/2)\leq(1+\cos\theta)/2+o(1)$ for all $x\in
B_{\sigma}\cap\mathrm{spt}\sigma_{V_{i},g_{i}}\cap\mathbb{R}^{n}$ as
$i\to\infty$, and so by the monotonicity formula (2.62) and (4.8) we have

$\Theta_{V_{i}}(y_{i},r)\leq(1+\cos\theta)/2+o(1)\mbox{ for all
}r\in(0,(1-\sigma)/2).$ (4.10)

For sufficiently large $i$, without loss of generality we can assume that each
$r_{i}$ is taken so that, for every $r\in(r_{i},(1-\sigma)/2)$, we have that

$\mathcal{D}((V_{i}-y_{i})/r,q_{i,r}(V^{(\theta)}))\leq\varepsilon$ (4.11)

for some $q_{i,r}\in O(n)$. Consider the rescaled varifolds
$V_{i}^{\prime}=(V_{i}-y_{i})/r_{i}$, and note
$H^{tan}_{V_{i}^{\prime},g_{i}^{\prime}}\to 0$ in $L^{\infty}_{loc}$,
$g_{i}^{\prime}\to g_{eucl}$ in $C^{1}_{loc}$, and $\beta_{i}\to\theta$ in
$C^{1}_{loc}$. By Lemma 3.12, up to a subsequence we get convergence
$V_{i}^{\prime}\to V^{\prime}$ to some $(\theta,S^{\prime})$-capillary
varifold $V^{\prime}$ in $\mathbb{R}^{n+1}$. We also have $V_{i,I}^{\prime}\to
V_{I}^{\prime}$ and $V_{i,B}^{\prime}\to
V_{B}^{\prime}=\cos\theta[S^{\prime}]$ (for $S^{\prime}$ a Caccioppoli set in
$\\{x_{1}=0\\}$) as varifolds,
$\sigma_{V_{i}^{\prime},g_{i}^{\prime}}\to\sigma_{V^{\prime}}$ as Radon
measures, and $0\in\mathrm{spt}\sigma_{V^{\prime}}$. Then (4.10) implies that
the density ratio satisfies

$\Theta_{V^{\prime}}(0,\infty)\leq\frac{1+\cos\theta}{2}.$ (4.12)

This, by monotonicity, implies
$\Theta_{V^{\prime}}(x)\leq\frac{1+\cos\theta}{2}$ for any
$x\in\mathbb{R}^{n}\cap B_{1}$. Passing to further subsequence as necessary,
by (4.11) we can assume that there are Euclidean isometries $q_{r}\in O(n)$
for all $r\in(1,\infty)$ so that

$\mathcal{D}(V^{\prime}/r,q_{r}(V^{(\theta)}))\leq\varepsilon,\quad\forall
r>1.$

By Lemma 3.14 (ensuring $\varepsilon(n,\theta)$ is small) we have that
$0<\mathcal{H}^{n}(S^{\prime}\cap B_{1})<\omega_{n}$, and hence
$\partial^{*}S^{\prime}\cap B_{2}\neq\emptyset$. Take
$x\in\partial^{*}S^{\prime}\cap B_{2}$. From Lemma 3.12 and Lemma 4.2, we
conclude that any tangent cone to $V^{\prime}$ at $x$ must be
$q(V^{(\theta)})$ for some Euclidean isometry $q$. Particularly, we have that

$\theta_{V^{\prime}}(x)=\frac{1+\cos\theta}{2}$ (4.13)

By the standard monotonicity formula, (4.12) and (4.13) together imply that
$V^{\prime}=x+q(V^{(\theta)})$ for some point $x\in\\{x_{1}=0\\}$ and $q\in
O(n)$. Since $0\in\mathrm{spt}\sigma_{V^{\prime}}$, we must have that
$V^{\prime}=q(V^{(\theta)})$. We therefore must have

$\mathcal{D}((V_{i}-y_{i})/r,q(V^{(\theta)}))<\varepsilon\quad\forall
r\in(r_{i}/2,2r_{i}),$

for all $i\gg 1$, contradicting our choice of $r_{i}$. ∎

To proceed, for $\theta\in(0,\frac{\pi}{2})$, we define
$\Omega^{\theta}=\\{x_{1}\cos\theta+x_{n+1}\sin\theta<0,x_{1}<0\\}$. In the
next Lemma, we work with conical free-boundary stationary varifolds in the
Euclidean half space.

######  Lemma 4.5 (Boundary behavior at touching points).

Given $\theta_{0}>0$, there is an $\varepsilon(n,\theta_{0})$ with the
following property. Let $\theta\in(\theta_{0},\pi/2-\theta_{0})$ and let $V$
be a conical, stationary $(\theta,S)$-capillary varifold in $\mathbb{R}^{n+1}$
satisfying

$\displaystyle\inf_{q\in
O(\\{x_{1}=0\\})}\mathcal{D}(V,q(V^{(\theta)}))\leq\varepsilon$ (4.14)

  1. (1)

If there exists $\theta^{\prime}$ such that

$\mathrm{spt}\mu_{V}\subset\overline{\Omega^{\theta^{\prime}}},$ (4.15)

then $\theta^{\prime}\leq\theta$;

  2. (2)

If there exists $\theta^{\prime\prime}$ such that

$\mathrm{spt}\mu_{V_{I}}\subset\mathbb{R}^{n+1}\setminus\Omega^{\theta^{\prime\prime}},$
(4.16)

then $\theta^{\prime\prime}\geq\theta$.

###### Proof.

Throughout the proof, $\Psi(\varepsilon)$ denotes some non-decreasing function
(which may change from line to line), depending on $n$ and $\theta$, such that
$\Psi(0)=0$.

First note that since $V,V^{(\theta)}$ are conical, (4.14) implies
$\theta_{V}(0)\leq(1+\cos\theta)/2+\Psi(\varepsilon)$. By contradiction, let
us suppose that $V$ satisfies (4.15) for some $\theta^{\prime}>\theta$ or
(4.16) for some $\theta^{\prime\prime}<\theta$.

We argue by induction. Assume $n=1$. Then any conical, stationary
$(\theta,B)$-capillary varifold $V$ is a union of rays extending from the
origin and, by (4.14),
$\Theta_{V}(0,1)\leq(1+\cos\theta)/2+\Psi(\varepsilon)$. Therefore $V$ must be
one of: $0,\cos\theta[\mathbb{R}^{n}],V^{(\pi/2)},V^{(\theta)}$ or
$q(V^{(\theta)})$ for $q(x_{1},x_{2})=(x_{1},-x_{2})$. However all of these
possibilities are precluded by (4.14), Lemma 3.14 (ensuring
$\varepsilon(\theta)$ is sufficiently small) and either (4.15) if
$\theta^{\prime}>\theta$ or (4.16) if $\theta^{\prime\prime}<\theta$.

Suppose now by inductive hypothesis the Theorem holds for $n-1$ in place of
$n$. By Lemma 4.4, we have $\inf_{q\in
O(n)}\mathcal{D}((V-y)/r,q(V^{(\theta)}))\leq\Psi(\varepsilon)$ for all
$y\in\mathrm{spt}\sigma_{V}$ and all $r>0$. If we pick
$y\in\mathrm{spt}\sigma_{V}\cap\\{x_{1}=x_{n+1}=0\\}\setminus\\{0\\}$, then
any tangent cone $V^{\prime}$ to $V$ at $y$ splits off a line
$V^{\prime}=[\mathbb{R}]\times V^{\prime\prime}$ with $V^{\prime\prime}$ being
a $(\theta,B^{\prime\prime})$-capillary varifold satisfying

$\displaystyle\inf_{q\in
O(n)}\mathcal{D}(V^{\prime\prime},q(V^{(\theta)}))\leq\Psi(\varepsilon),$
$\displaystyle\mathrm{spt}\mu_{V^{\prime\prime}}\subset\overline{\Omega^{\theta^{\prime}}}\quad\text{or}\quad\mathrm{spt}\mu_{V^{\prime\prime}_{I}}\subset\mathbb{R}^{n+1}\setminus\Omega^{\theta^{\prime\prime}}.$

By inductive hypothesis if we ensure $\varepsilon(n,\theta)$ is sufficiently
small then $V^{\prime\prime}$ cannot exist, and therefore we must have
$\mathrm{spt}\sigma_{V}\cap\\{x_{1}=x_{n+1}=0\\}\subset\\{0\\}$. By Theorem
3.1 we deduce

$\displaystyle\mathrm{spt}\mu_{V_{I}}\cap\\{x_{1}=x_{n+1}=0\\}\subset\\{0\\}$
(4.17)

also.

By increasing $\theta^{\prime}$ (resp. decreasing $\theta^{\prime\prime}$) we
can assume that $\theta^{\prime}$ is the largest angle for which
$\mathrm{spt}\mu_{V}\subset\overline{\Omega^{\theta^{\prime}}}$ (resp.
$\theta^{\prime\prime}$ is the smallest angle for which
$\mathrm{spt}\mu_{V_{I}}\subset\mathbb{R}^{n+1}\setminus\Omega^{\theta^{\prime\prime}}$).
Assume that $V$ satisfies (4.15) for some $\theta^{\prime}>\theta$: for the
remainder of the proof, the argument in the other case is identical. Since
$\mathrm{spt}\mu_{V_{I}}$ is a closed cone, there exists
$y\in\mathrm{spt}\mu_{V_{I}}\cap P_{\theta^{\prime}}^{-}\cap\partial B_{1}$.
However, by (4.17), it must hold that $y_{1}<0$. Therefore near $y$, $V_{I}$
is stationary and, since $V$ is conical, $\mathrm{spt}\mu_{V_{I}}$ touches
$P_{\theta^{\prime}}^{-}$ from one side. By the strong maximum principle [25]
we deduce $P_{\theta^{\prime}}^{-}\subset\mathrm{spt}\mu_{V_{I}}$. But this
means
$\mathrm{spt}\mu_{V_{I}}\cap\\{x_{1}=x_{n+1}=0\\}=\\{x_{1}=x_{n+1}=0\\}$,
contradicting (4.17). This proves the Lemma with $n$ in place of $n-1$, and so
by induction the Lemma holds for all $n$. ∎

We are now ready to establish the boundary maximum principle.

###### Proof of Theorem 4.1.

Denote by $\theta^{\prime}$ the $g$-angle between $G_{\theta}(u)$ and
$\\{x_{1}=0\\}$ at $z$, i.e. so
$\cos\theta^{\prime}=g(\nu_{\theta}(u,g)|_{z},\nu_{g}|_{z})$. Write
$\Omega_{\theta}(u)=\\{(x_{1},\ldots,x_{n+1})\in C^{\theta}_{1/2}\cap
B_{3/4}:x_{n+1}<-x_{1}\cot\theta+u(x^{\prime})\\}$ for the subgraph of $u$ in
$B_{3/4}$, and note that up to rotation the tangent cone of
$\Omega_{\theta}(u)\cap\\{x_{1}\leq 0\\}$ at $z$ is
$\Omega^{\theta^{\prime}}=\\{x_{1}\cos\theta^{\prime}+x_{n+1}\sin\theta^{\prime}<0,x_{1}\leq
0\\}$.

Let us first consider Case 1. We claim that

$\mathrm{spt}\mu_{V}\cap C^{\theta}_{1/2}\cap
B_{3/4}\subset\overline{\Omega_{\theta}(u)}$ (4.18)

if $\delta(n,\theta)$ is chosen sufficiently small. To see this, first note
that for $\delta(n,\theta)$ small (4.1) implies $\Theta_{V}(x,1/8)\leq
3/4+\cos\theta/4$ for all $x\in B_{3/4}\cap\mathbb{R}^{n}$. Theorem 3.1 then
gives $\mu_{V_{B}}\llcorner B_{3/4}=\cos\beta[S]_{g}$ for some set
$S\subset\mathbb{R}^{n}$ satisfying

$\partial S\cap B_{3/4}\subset\mathrm{spt}\sigma_{V,g}\cap
B_{3/4}=\mathrm{spt}\mu_{V_{I}}\cap\mathbb{R}^{n}\cap B_{3/4}.$ (4.19)

From Lemma 3.12, after shrinking $\delta(n,\theta)$ as necessary, we have

$\mathrm{spt}\mu_{V_{I}}\cap B_{3/4}\subset B_{\cos\theta/100}(P_{\theta}),$

and (therefore, in combination with (4.19)), we have

$\\{x_{1}=0,x_{n+1}<-1/100\\}\subset
S\subset\\{x_{1}=0,x_{n+1}<1/100\\}\quad\text{ in }B_{3/4}.$ (4.20)

(4.19), (4.20) together imply that for any
$(x^{\prime},x_{n+1})\in\mathrm{spt}\mu_{V_{B}}\cap C^{\theta}_{1/2}\cap
B_{3/4}$, there is a
$(x^{\prime},y_{n+1})\in\mathrm{spt}\mu_{V_{I}}\cap\mathbb{R}^{n}\cap
C^{\theta}_{1/2}\cap B_{3/4}$ with $x_{n+1}\leq y_{n+1}$. In other words,
$\mathrm{spt}\mu_{V_{B}}$ lies below $\mathrm{spt}\mu_{V_{I}}$ in
$C^{\theta}_{1/2}\cap B_{3/4}$, which in turn lies below $G_{\theta}(u)$,
giving us our claim.

Now from Lemma 4.4 (provided $\delta(n,\theta)$ is small) any tangent cone
$V^{\prime}$ to $V$ at $z$ is non-zero (since
$z\in\mathrm{spt}V_{I}\cap\mathbb{R}^{n}$) and satisfies

$\inf_{q\in O(n)}\mathcal{D}(V^{\prime},q(V^{(\beta(z))}))<\varepsilon_{0},$

where $\varepsilon_{0}$ is the constant from Lemma 4.5. On the other hand,
from (4.18), $V^{\prime}$ must be supported in
$\overline{\Omega^{\theta^{\prime}}}$, thus by Lemma 4.5 we must have
$\theta^{\prime}\leq\beta(z)$. This proves Case 1.

Case 2 is similar but easier. In this case we clearly have
$\mathrm{spt}\mu_{V_{I}}\cap C^{\theta}_{1/2}\cap B_{3/4}\subset
C^{\theta}_{1/2}\cap B_{3/4}\setminus\Omega_{\theta}(u)$, and therefore any
tangent cone $V^{\prime}$ to $V$ at $z$ satisfies
$\mathrm{spt}\mu_{V_{I}^{\prime}}\subset\mathbb{R}^{n+1}\setminus\Omega_{\theta^{\prime}}$.
Lemma 4.5 implies $\theta^{\prime}\geq\beta(z)$, proving Case 2. ∎

##  5\. Partial harnack

In the next two sections we will make the following running assumptions on a
varifold $V$.

######  Assumption 5.1.

We assume $\theta_{0}\in(0,\pi/2)$, $g$ is a $C^{1}$ metric in $B_{1}$,
$\beta$ a $C^{1}$ function on $B_{1}$, and $V$ is $(\beta,S)$-capillary
varifold in $(B_{1},g)$ satisfying

$\displaystyle 0\in\mathrm{spt}{V_{I}},\quad\Theta_{V}(0,1)\leq
3/4+\cos\theta_{0}/4<1,$ (5.1)

Our main result in this section is Theorem 5.3, the partial Harnack
inequality. This will tell us that inhomogenous blow-ups of a sequence of
varifolds $V_{i}\to V^{(\theta)}$ will converge in support to some
$C^{\alpha}$ function. The proof relies on the strategy of [9] (c.f. [28]),
originally implemented to study the regularity of the one-phase Bernoulli
problem. We first establish the decay of oscillation.

######  Theorem 5.2.

There is a $\gamma(n,\theta_{0})\in(0,1/100)$ so that the following holds. If
$V$ satisfies Assumptions 5.1 and additionally

$\displaystyle\max\\{\mathrm{osc}_{P_{\beta(0)}}(V_{I},B_{1}),|\beta(0)-\theta_{0}|\\}\leq\gamma,$
(5.2)
$\displaystyle\max\\{||H^{tan}_{V,g}||_{L^{\infty}(B_{1})},|D\beta|_{C^{0}(B_{1})},|g-g_{eucl}|_{C^{1}(B_{1})}\\}\leq\gamma\mathrm{osc}_{P_{\beta(0)}}(V_{I},B_{1}),$
(5.3)

then

$\mathrm{osc}_{P_{\beta(0)}}(V_{I},B_{\gamma})\leq(1-\gamma)\mathrm{osc}_{P_{\beta(0)}}(V_{I},B_{1}).$
(5.4)

###### Proof.

We first introduce some notation. Let $\theta=\beta(0)$. Given
$x^{\prime}\in{P_{\pi/2}}$, define

$|x^{\prime}|_{\theta}:=\sqrt{(\sin\theta)^{-2}x_{1}^{2}+x_{2}^{2}+\ldots
x_{n}^{2}}.$

Observe that $B^{\theta}_{r}(x_{0}^{\prime})=\\{x^{\prime}\in
P_{\pi/2}:|x^{\prime}-x_{0}^{\prime}|_{\theta}<r\\}$, and by a direct
computation:

$\mathcal{L}_{\theta}(|x^{\prime}|^{-n}_{\theta})=2n|x^{\prime}|^{-n-2}_{\theta},$

where recall that $\mathcal{L}_{\theta}(u)=\Delta u-\cos^{2}\theta
D_{1}^{2}u$.

Fix $x_{0}^{\prime}=-(\sin\theta/4)e_{1}$, and write
$\eta=\mathrm{osc}_{P_{\theta}}(V_{I},B_{1})$. After translating $V$ there is
no loss in assuming that for all $x\in\mathrm{spt}V_{I}\cap B_{19/20}$ we have
$|x\cdot\nu_{\theta}|\leq\eta$ (recall $\nu_{\theta}=\cos\theta
e_{1}+\sin\theta e_{n+1}$), and hence $|x_{n+1}+\cot\theta
x_{1}|\leq\eta/\sin\theta$. Provided $\eta,\gamma$ are sufficiently small
(depending only on $n,\theta_{0}$), Allard’s theorem implies that

$\mathrm{spt}V_{I}\cap C^{\theta}_{1/8}(x_{0}^{\prime})\cap
B_{9/10}=G_{\theta}(w)$

for some $C^{1,1/2}$ function
$w:B^{\theta}_{1/8}(x_{0}^{\prime})\to\mathbb{R}$ satisfying

$|w|_{C^{1,1/2}}\leq c(n,\theta_{0})\eta,\quad|w|\leq\eta/\sin\theta.$ (5.5)

Suppose $w(x_{0}^{\prime})\leq 0$. Then by (5.2), (5.5) there is a radius
$r_{0}(n,\theta_{0})$ (which we now fix) so that

$w\leq\frac{\eta}{2\sin\theta}\text{ on }B^{\theta}_{2r_{0}}(x_{0}^{\prime}).$
(5.6)

For $t\in\mathbb{R}$, define $u_{t}:{P_{\pi/2}}\to\mathbb{R}$ by

$u_{t}(x^{\prime})=\left\\{\begin{array}[]{l
l}t&1/2\leq|x^{\prime}-x_{0}^{\prime}|_{\theta}\\\
t-\eta\bar{c}(|x^{\prime}-x_{0}^{\prime}|^{-n}_{\theta}-2^{n})&r_{0}\leq|x^{\prime}-x_{0}^{\prime}|_{\theta}\leq
1/2\\\
t-\eta\bar{c}(r_{0}^{-n}-2^{n})&|x^{\prime}-x_{0}^{\prime}|_{\theta}\leq
r_{0}\end{array}\right.,$

for $\bar{c}=\frac{1}{4\sin\theta}(r_{0}^{-n}-2^{n})^{-1}$, so that $u_{t}$ is
continuous and

$u_{t}=t\text{ outside }B^{\theta}_{1/2}(x_{0}^{\prime})\quad\text{and}\quad
u_{t}=t-\frac{\eta}{4\sin\theta}\text{ inside
}B^{\theta}_{r_{0}}(x_{0}^{\prime}).$ (5.7)

We will use $G_{\theta}(u_{t})$ as barrier hypersurfaces for $V_{I}$.

For $x^{\prime}\in B^{\theta}_{1/2}(x^{\prime}_{0})\setminus
B^{\theta}_{r_{0}}(x^{\prime}_{0})$, we compute

$H_{\theta}(u_{t},g)|_{x^{\prime}}\leq-2n\eta\bar{c}|x^{\prime}-x^{\prime}_{0}|_{\theta}^{-n-2}+c(n,\theta_{0})(\eta+\gamma)\leq-c(n,\theta_{0})\eta$
(5.8)

provided $\gamma(n,\theta_{0})$ is sufficiently small. For $x^{\prime}\in
B^{\theta}_{1/2}(x^{\prime}_{0})\cap\\{x_{1}=0\\}$ we compute

$\displaystyle\nu_{\theta}(u_{t},g)|_{x^{\prime}}\cdot e_{1}$
$\displaystyle\leq\cos\theta+\frac{\eta\bar{c}n}{4\sin\theta}|x^{\prime}-x_{0}^{\prime}|_{\theta}^{-n-2}(\cos^{2}\theta\sin\theta-1)+c(n,\theta_{0})\gamma\eta$
$\displaystyle\leq\cos\theta-\eta\bar{c}\sin\theta$ (5.9)

again provided $\gamma(n,\theta_{0})$ is small.

We claim that $\mathrm{spt}V_{I}\cap B_{9/10}$ must lie below
$G_{\theta}(u_{t_{0}})$ with $t_{0}=\eta/\sin\theta$. In other words we claim
that for every $(x^{\prime},x_{n+1})\in\mathrm{spt}V_{I}\cap B_{9/10}$ we have
$x_{n+1}\leq-\cot\theta x_{1}+u_{t_{0}}(x^{\prime})$. To see this, we first
observe by (5.5), (5.7) that $\mathrm{spt}V_{I}\cap B_{9/10}$ lies below
$G_{\theta}(u_{t_{1}})$ with $t_{1}=2\eta/\sin\theta$, and we second observe
that both $\mathrm{spt}V_{I}\cap B_{9/10}\cap
C^{\theta}_{2r_{0}}(x_{0}^{\prime})$ (by (5.6)) and $\mathrm{spt}V_{I}\cap
B_{9/10}\setminus C^{\theta}_{1/2}(x_{0}^{\prime})$ lie below
$G_{\theta}(u_{t_{0}})$.

Now consider the smallest $t_{*}\in[t_{0},t_{1}]$ for which
$\mathrm{spt}V_{I}\cap B_{9/10}$ lies below $G_{\theta}(u_{t_{*}})$, and
suppose towards a contradiction that $t_{*}>t_{0}$. Then by the previous
paragraph and (5.5) there must be a point $x_{*}\in\mathrm{spt}V_{I}\cap
G_{\theta}(u_{t_{*}})\cap\overline{C^{\theta}_{1/2}(x^{\prime}_{0})}\setminus
C^{\theta}_{2r_{0}}(x^{\prime}_{0})$. By the strong maximum principle, (5.8),
and our assumption (5.3) (provided $\gamma(n,\theta_{0})$ is small),
$x_{*}\not\in\\{x_{1}<0\\}$. On the other hand, taking $\gamma(n,\theta_{0})$
small, from Lemma 3.15 and Theorem 4.1 (applied to the function
$x^{\prime}\mapsto
t_{*}-\eta\bar{c}(|x^{\prime}-x^{\prime}_{0}|_{\theta}^{-n}-2^{n})$) if
$x_{*}\in\\{x_{1}=0\\}$ then we would have

$\nu_{\theta}(u_{t_{*}},g)|_{x^{\prime}}\cdot
e_{1}\geq\cos\beta(x_{*})\geq\cos\theta-\gamma\eta$

which contradicts (5.9) for $\gamma(n,\theta_{0})$ small. So $x_{*}$ cannot
exist, and we must have $t_{*}=t_{0}$, proving our claim.

To conclude, we observe that for $x^{\prime}\in B^{\theta}_{1/8}(0)\subset
B^{\theta}_{1/2}(x^{\prime}_{0})$ we have
$|x^{\prime}-x^{\prime}_{0}|^{-n}_{\theta}-2^{n}\geq 2^{n}/3$, and so

$u_{t_{0}}\leq t_{0}-\bar{c}\eta/2\leq(1-\gamma)\eta/\sin\theta\quad\text{ on
}\quad B^{\theta}_{1/8}(0).$

Ensuring $\eta(n,\theta_{0})$ is small, we deduce that
$\mathrm{osc}_{P_{\theta}}(V_{I},B_{1/16})\leq(1-\gamma)\mathrm{osc}_{P_{\theta}}(V_{I},B_{1})$.
This proves the Theorem in the case when $w(x^{\prime}_{0})\leq 0$. If
$w(x^{\prime}_{0})\geq 0$ then we can use an essentially verbatim argument,
replacing $-\eta$ with $\eta$ in the definition of $u_{t}$ and touching
$\mathrm{spt}V_{I}\cap B_{9/10}$ from below by $G_{\theta}(u_{t})$. ∎

######  Theorem 5.3 (Partial Harnack).

Given $\tau>0$, there are $c(n,\theta_{0})$, $\varepsilon(n,\theta_{0},\tau)$,
$\alpha(n,\theta_{0})$ so that the following holds. If $V$ satisfies
Assumption 5.1 and additionally

$\displaystyle
E:=\max\\{\mathrm{osc}_{P_{\beta(0)}}(V_{I},B_{1}),||H^{tan}_{V,g}||_{L^{\infty}(B_{1})},|D\beta|_{C^{0}(B_{1})},|g-g_{eucl}|_{C^{1}(B_{1})}\\}\leq\varepsilon,$
(5.10) $\displaystyle|\beta(0)-\theta_{0}|\leq\varepsilon,$ (5.11)

then for all $x\in\mathrm{spt}V_{I}\cap B_{1/2}$ and all $\tau\leq r\leq 1/2$,
we have

$\displaystyle\mathrm{osc}_{P_{\beta(0)}}(V_{I},B_{r}(x))\leq cr^{\alpha}E.$
(5.12)

###### Proof.

We use Theorem 5.2 to get Hölder decay centered at boundary points, and
Allard’s theorem to get Hölder decay at interior points up until the radius is
comparable to $|x_{1}|$. We combine the two decays to get decay for all radii
$\geq\tau$. There is no loss in assuming $\tau\leq 1/100$. For ease of
notation write $\theta=\beta(0)$.

First let $\gamma$ be the constant from Theorem 5.2, and choose $K\geq
4/\gamma$. For $x\in\mathrm{spt}V_{I}\cap\\{x_{1}=0\\}\cap
B_{1/2}\equiv\mathrm{spt}\sigma_{V,g}\cap B_{1/2}$, define the function

$F(r)=\mathrm{osc}_{P_{\beta(x)}}(V_{I},B_{r}(x))+KEr.$

We claim that

$F(\gamma r)\leq(1-\gamma)F(r)\quad\text{for all $r$ such
that}\quad\mathrm{osc}_{P_{\beta(x)}}(V_{I},B_{1})/\gamma\leq r\leq 1/4$
(5.13)

To see this, note that for such $r$ we trivially have
$\mathrm{osc}_{P_{\beta(x)}}(V_{I},B_{r}(x))/r\leq\gamma$. If, in addition, we
have

$\displaystyle\max\\{r||H^{tan}_{V,g}||_{L^{\infty}(B_{r}(x))},r|D\beta|_{C^{0}(B_{r}(x))},|g-g_{eucl}|_{C^{0}(B_{r}(x))},r|Dg|_{C^{0}(B_{r}(x))}\\}$
$\displaystyle\quad\leq\gamma\mathrm{osc}_{P_{\beta(x)}}(V_{I},B_{r}(x))/r,$
(5.14)

then we can apply Theorem 5.2 at scale $B_{r}(x)$ to get

$F(\gamma r)\leq(1-\gamma)\mathrm{osc}_{P_{\beta(x)}}(V_{I},B_{r}(x))+KE\gamma
r\leq(1-\gamma)F(r)$

On the other hand, if (5.14) fails, then we can compute instead

$\displaystyle F(\gamma r)$ $\displaystyle\leq
KEr\left(\frac{1}{K\gamma}+\gamma\right)\leq(1-\gamma)F(r)$

This proves (5.13). By iterating (5.13), and ensuring
$\varepsilon(n,\theta_{0},\tau)$ is small, we deduce that with the choice
$\alpha=\log(1-\gamma)/\log(\gamma)\in(0,1)$ it holds

$\mathrm{osc}_{P_{\theta}}(V_{I},B_{r}(x))\leq\mathrm{osc}_{P_{\beta(x)}}(V_{I},B_{r}(x))+c(n)r|D\beta|_{C^{0}(B_{1})}\leq
c(n,\theta_{0})r^{\alpha}E$ (5.15)

for all $x\in\mathrm{spt}V_{I}\cap\\{x_{1}=0\\}\cap B_{1/2}$ and all $\tau\leq
r\leq 1/2$, provided we ensure $\varepsilon(n,\tau)$ is small enough to
guarantee

$\mathrm{osc}_{\beta(x)}(V_{I},B_{1})\leq\mathrm{osc}_{\beta(0)}(V_{I},B_{1})+c(n)|D\beta|_{C^{0}(B_{1})}\leq
c(n)\varepsilon\leq\tau.$

Second, since $V\to V^{(\theta_{0})}$ as $\varepsilon\to 0$ (by Lemma 3.15)
and $\theta\to\theta_{0}$, then by Allard’s theorem and provided
$\varepsilon(n,\theta_{0},\tau)$ is sufficiently small we can write

$\mathrm{spt}V_{I}\cap C^{\theta}_{9/10}\cap\\{x_{1}<-\tau\\}=G_{\theta}(w),$

for $w:B^{\theta}_{9/10}\cap\\{x_{1}<-\tau\\}\to\mathbb{R}$ a $C^{1}$ function
satisfying the following estimate: for any $x=(x^{\prime},x_{n+1})\in
G_{\theta}(w)$, we have

$\sup_{B^{\theta}_{|x_{1}|/4}(x^{\prime})}\left(|x_{1}|^{-1}|w|+|Dw|\right)\\\
\leq
c\left(\frac{\mathrm{osc}_{P_{\theta}}(V_{I},B_{|x_{1}|}(x))}{|x_{1}|}+|x_{1}|||H_{V,g}^{tan}||_{L^{\infty}(B_{|x_{1}|}(x))}+|x_{1}||Dg|_{C^{0}(B_{|x_{1}|}(x))}\right)$
(5.16)

for some $c(n,\theta_{0})$. Therefore (using that $\alpha\in(0,1)$) we deduce
that

$\mathrm{osc}_{P_{\theta}}(V_{I},B_{r}(x))\leq
c\left(\frac{r}{|x_{1}|}\right)^{\alpha}\Big{(}\mathrm{osc}_{P_{\theta}}(V_{I},B_{|x_{1}|}(x))+E|x_{1}|^{2}\Big{)}$
(5.17)

for all $x\in\mathrm{spt}V_{I}\cap B_{1/2}\cap\\{x_{1}<-\tau\\}$ and all
$0<r<|x_{1}|$, where $c=c(n,\theta_{0})$.

Thirdly, again ensuring $\varepsilon(n,\theta_{0},\tau)$ is small, by Lemma
3.12 we can assume that

$d_{H}(\mathrm{spt}V_{I}\cap\\{x_{1}=0\\}\cap
B_{9/10},\\{x_{1}=x_{n+1}=0\\}\cap B_{9/10})\leq\tau.$ (5.18)

Take $x\in\mathrm{spt}V_{I}\cap B_{1/2}$. From (5.18) and the assumption
$\mathrm{osc}_{P_{\beta(0)}}(V_{I},B_{1})\leq\varepsilon\leq\tau$, we can
choose $y\in\mathrm{spt}V_{I}\cap\\{x_{1}=0\\}\cap B_{1/2}$ so that
$|x-y|\leq|x_{1}|/\sin\beta(0)+2\tau\leq c(\theta_{0})|x_{1}|+2\tau$, where
the latter inequality holds true provided $\varepsilon$ is small enough
depending on $\theta_{0}$, since $|\beta-\theta_{0}|_{C^{0}}\leq\varepsilon$.

For $r\geq\max(\tau,|x_{1}|)$, we can use (5.15) and our choice of $y$ to
estimate

$\mathrm{osc}_{P_{\theta}}(V_{I},B_{r}(x))\leq\mathrm{osc}_{P_{\theta}}(V_{I},B_{r+|x-y|}(y))\leq
c(r+|x-y|)^{\alpha}E\leq c(n,\theta_{0})r^{\alpha}E,$

while for $\tau\leq r\leq|x_{1}|$, it holds $|x-y|\leq c(\theta_{0})|x_{1}|$,
thus by (5.17), (5.15) we get

$\displaystyle\mathrm{osc}_{P_{\theta}}(V_{I},B_{r}(x))$ $\displaystyle\leq
c(r/|x_{1}|)^{\alpha}\mathrm{osc}_{P_{\theta}}(V_{I},B_{|x_{1}|+|x-y|}(y))+cr^{\alpha}E$
$\displaystyle\leq c(r/|x_{1}|)^{\alpha}|x_{1}|^{\alpha}E+cr^{\alpha}E\leq
c(n,\theta_{0})r^{\alpha}E.$

This completes the proof of Theorem 5.3. ∎

##  6\. Decay estimate

We continue to make the same Assumptions 5.1 on $V$ as in Section 5. The
following (Theorem 6.1) is the main regularity/excess decay estimate, and it
follows from an iteration of Theorem 6.3, proved below.

######  Theorem 6.1 (Excess decay estimate).

There are $\varepsilon(n,\theta_{0})$, $c(n,\theta_{0})$ positive so that if
$V$ satisfies Assumptions 5.1 and additionally satisfies

$E:=\max\\{\mathrm{osc}_{P_{\theta_{0}}}(V_{I},B_{1}),||H^{tan}_{V,g}||_{L^{\infty}(B_{1})},|D\beta|_{C^{0}(B_{1})},|g-g_{eucl}|_{C^{1}(B_{1})}\\}\leq\varepsilon,$
(6.1)

then for all $x\in\mathrm{spt}V_{I}\cap B_{1/8}$, we can find an
$n$-dimensional plane $Q_{x}$ with $|Q_{x}-P_{\theta_{0}}|\leq cE$ so that for
$0<r<1/2$ we have

$\mathrm{osc}_{Q_{x}}(V_{I},B_{r}(x))\leq cr^{3/2}E.$ (6.2)

######  Remark 6.2.

Of course there is nothing special in the power $3/2$, any number in $(1,2)$
would equally work.

###### Proof of Theorem 6.1.

For ease of notation write $\theta=\theta_{0}$. By Lemma 3.15, as
$\varepsilon\to 0$ we have $V\to V^{(\theta)}$ in $B_{1}$ and $\beta\to\theta$
in $C^{0}(B_{1})$, and so there is no loss in assuming that

$\displaystyle\Theta_{V}(x,1/2)\leq(1+\cos\theta)/2+\varepsilon^{\prime}\quad\forall
x\in\mathrm{spt}V_{I}\cap\\{x_{1}=0\\}\cap B_{1/4},$ (6.3)
$\displaystyle|\beta-\theta|_{C^{0}(B_{1})}\leq\varepsilon^{\prime}$ (6.4)

for any particular $\varepsilon^{\prime}(n,\theta)$ which we shall fix later.

_Step 1: Decay towards Euclidean metric on the boundary._ Fix
$x\in\mathrm{spt}V_{I}\cap\\{x_{1}=0\\}\cap B_{1/4}$, and suppose
$g(x)=g_{eucl}$. Choose $\eta(n,\theta)$ so that $c\eta^{2}\leq\eta^{3/2}$
where $c(n,\theta)$ is the constant from Theorem 6.3, and let
$\varepsilon_{1}(n,\theta,\eta)$ be the corresponding $\varepsilon$. For a
$n$-plane $Q$, $0<r<1$, define

$\displaystyle E(Q,x,r)$
$\displaystyle=\max\\{r^{-1}\mathrm{osc}_{Q}(V_{I},B_{r}(x)),\varepsilon_{1}^{-1}r||H_{V,g}||_{L^{\infty}(B_{r}(x))},$
$\displaystyle\quad\quad\varepsilon_{1}^{-1}r|D\beta|_{C^{0}(B_{r}(x))},\varepsilon_{1}^{-1}r|Dg|_{C^{0}(B_{r}(x))}\\},$

We claim we can find a $Q_{x}$ so that, for all $0<r<1/4$,

$|Q_{x}-P_{\beta(x)}|\leq cE(P_{\beta(x)},x,1/4),\quad E(Q_{x},x,r)\leq
cr^{1/2}E(P_{\beta(x)},x,1/4),$ (6.5)

for $c=c(n,\theta)$.

Let us prove our claim. First note that by (6.3), monotonicity (2.62), and our
assumptions on $g$, provided $\varepsilon^{\prime}(n,\theta)$ is chosen
sufficiently small we have

$\displaystyle\Theta_{V}(x,r)\leq 3/4+\cos\theta/4\quad\forall 0<r<1/2,$ (6.6)
$\displaystyle|\beta(x)-\theta_{0}|\leq\varepsilon_{1}.$ (6.7)

So, for any particular $0<r<1/2$, if we define

$\displaystyle V_{x,r}=(T_{x,r})_{\sharp}V,\quad
g_{x,r}(y)=g(x+ry),\quad\beta_{x,r}(y)=\beta(x+ry),$ (6.8)
$\displaystyle\text{ for }T_{x,r}(y)=(y-x)/r:(B_{r}(x),g)\to(B_{1},g_{x,r}),$
(6.9)

(note that $g_{x,r}$ is chosen so that $JT_{x,r}=r^{-n}$ on $V$) then
$V_{x,r}$ as a varifold in $(B_{1},g_{x,r})$ will also satisfy Assumption 5.1,
with $g_{x,r}$, $\beta_{x,r}$ in place of $g,\beta$. Moreover, since
$g_{x,r}(0)=g_{eucl}$, we have

$|g_{x,r}-g_{eucl}|_{C^{0}(B_{1})}\leq c(n)r|Dg|_{C^{0}(B_{r}(x))}.$

For $i\in\\{0,1,2,\ldots\\}$, set $r_{i}=\eta^{i}/4$, and define
$Q_{0}=P_{\beta(x)}$, $E_{0}=E(Q_{0},x,1/4)\equiv E(Q_{0},x,r_{0})\leq c(n)E$.
In light of the previous discussion, ensuring
$\varepsilon(\varepsilon_{1},n,\theta)$ is sufficiently small, we can apply
Theorem 6.3 to $V_{x,r_{i}}$, $g_{x,r_{i}},\beta_{x,r_{i}}$, and inductively
define $Q_{i}$ so that

$\displaystyle\mathrm{osc}_{Q_{i+1}}(V_{I},B_{r}(x))\leq\eta^{3/2}E_{i},\quad|Q_{i+1}-Q_{i}|\leq
c(n,\theta)E_{i}$ (6.10)

where $E_{i}=E(Q_{i},x,r_{i})$. In particular, we have

$E_{i}\leq\eta^{i/2}E_{0},$

and so $Q_{i}\to Q_{x}$. Given $r\in(0,1/4)$, choose the greatest $i$ so that
$r\leq r_{i}$, and then we can estimate

$r^{-1}\mathrm{osc}_{Q_{x}}(V_{I},B_{r}(x))\leq cE_{i}+c|Q_{x}-Q_{i}|\leq
cr_{i}^{1/2}E_{0}\leq c(n,\theta)r^{1/2}E_{0},$

and $|Q_{x}-P_{\beta(x)}|\leq c(n,\theta)E_{0}$. Our required inequality (6.5)
follows from the above and the inequality $|D\beta|\leq E$.

_Step 2: Decay towards non-Euclidean metric on the boundary._ If
$x\in\mathrm{spt}V_{I}\cap\\{x_{1}=0\\}\cap B_{1/4}$ but $g(x)\neq g_{eucl}$,
then we can choose an affine map $L$ satisfying

$L(x)=x,\quad|DL-Id|\leq c|g(x)-g_{eucl}|\leq c(n)\varepsilon,\quad
L(\\{x_{1}\leq 0\\})=\\{x_{1}\leq 0\\},$ (6.11)

so that if $\tilde{g}=(L^{-1})^{*}g$ then $\tilde{g}(x)=g_{eucl}$. If we let
$\tilde{V}=L_{\sharp}V$ (where we view $L$ as an isometry
$(B_{1},g)\to(L(B_{1}),\tilde{g}))$ and $\tilde{\beta}=\beta\circ L^{-1}$,
then by (6.3), (2.62), (2.3) the translated/scaled $\tilde{V}_{x,1/4}$,
$\tilde{g}_{x,1/4}$, $\tilde{\beta}_{x,1/4}$ satisfy Assumption 5.1, and
hypotheses (6.1) with $c(n)\varepsilon$ in place of $\varepsilon$.

We can therefore apply Step 1 to deduce the existence of an $n$-plane
$\tilde{Q}_{x}$ for which

$|\tilde{Q}_{x}-P_{\theta}|\leq
c\tilde{E},\quad\mathrm{osc}_{\tilde{Q}_{x}}(\tilde{V}_{I},B_{r}(x))\leq
cr^{3/2}\tilde{E}\quad\forall 0<r<1/4,$ (6.12)

where $c(n,\theta)$ and

$\tilde{E}=\max\\{\mathrm{osc}_{P_{\theta}}(\tilde{V}_{I},B_{1/4}(x)),||H_{\tilde{V},\tilde{g}}||_{L^{\infty}(B_{1/4}(x))},|D\tilde{\beta}|_{C^{0}(B_{1/4}(x))},|D\tilde{g}|_{C^{0}(B_{1/4}(x))}\\}.$
(6.13)

Define $Q_{x}=L^{-1}\tilde{Q}_{x}$. Then from (2.3), (6.12), (6.13), ensuring
$\varepsilon(n,\theta)$ is small, we deduce for any $0<r<1/8$ the inequalities

$\mathrm{osc}_{Q_{x}}(V_{I},B_{r}(x))\leq
2\mathrm{osc}_{\tilde{Q}_{x}}(\tilde{V}_{I},B_{2r}(x))\leq
4cr^{3/2}\tilde{E}\leq 8cr^{3/2}E$

(for $c$ as in (6.5)), and

$|Q_{x}-P_{\theta}|\leq c(n)|g(x)-g_{eucl}|+|\tilde{Q}_{x}-P_{\theta}|\leq
c\tilde{E}+c(n)E\leq c(n,\theta)E.$

This completes Step 2, and implies (6.2) for any
$x\in\mathrm{spt}V_{I}\cap\\{x_{1}=0\\}\cap B_{1/4}$.

_Step 3: Decay away from the boundary._ We first claim that
$\mathrm{spt}V_{I}\cap\mathbb{R}^{n}\cap B_{1/8}$ is contained in the graph of
a $C^{1,1/2}$ function $u:\mathbb{R}^{n-1}\cap B_{1/8}\to\mathbb{R}$ with
$|u|_{C^{1,1/2}}\leq c(n,\theta)E$, where
$\mathbb{R}^{n-1}=\\{x_{1}=x_{n+1}=0\\}$.

To see this, write $\Sigma^{\prime}=\mathrm{spt}V_{I}\cap\mathbb{R}^{n}$. From
Step 2, for every $x\in\Sigma^{\prime}\cap B_{1/4}$ we have an $n$-dimensional
plane $Q_{x}$ satisfying $|Q_{x}-P_{\theta}|\leq c(n,\theta)E$ so that

$|\pi_{Q_{x}}^{\perp}(y-x)|\leq cE|x-y|^{3/2}\quad\forall
y\in\Sigma^{\prime}\cap B_{1/2}.$ (6.14)

For $\varepsilon(n)$ sufficiently small, we deduce there is a function
$u:\pi_{\mathbb{R}^{n-1}}(\Sigma^{\prime}\cap B_{1/4})\to\mathbb{R}$ so that

$\Sigma^{\prime}\cap
B_{1/4}=\\{(0,x^{\prime},u(x^{\prime})):x^{\prime}\in\pi_{\mathbb{R}^{n-1}}(\Sigma^{\prime}\cap
B_{1/4})\\},$

and with the property that for every
$x^{\prime},y^{\prime}\in\pi_{\mathbb{R}^{n-1}}(\Sigma^{\prime}\cap B_{1/4})$,
there is a linear function $\ell_{x^{\prime}}:\mathbb{R}^{n-1}\to\mathbb{R}$
so that

$|u(x^{\prime})|\leq
c(n,\theta)E,\quad|u(y^{\prime})-u(x^{\prime})-\ell_{x^{\prime}}(y^{\prime}-x^{\prime})|\leq
c(n,\theta)|y^{\prime}-x^{\prime}|^{3/2}E.$ (6.15)

On the other hand, for any $x\in\Sigma^{\prime}\cap B_{1/4}$ we have by
(6.14), Lemma 3.15, and Lemma 3.12 that

$\mathrm{spt}(V_{x,r,I})\cap B_{1}\equiv(\mathrm{spt}V_{I}-x)/r\cap B_{1}\to
Q_{x}\cap\\{x_{1}\leq 0\\}$

in the Hausdorff distance as $r\to 0$, and hence
$\pi_{\mathbb{R}^{n-1}}((\Sigma^{\prime}-x)/r)\to\mathbb{R}^{n-1}$ in the
local Hausdorff distance as $r\to 0$ also. We deduce we must have
$\pi_{\mathbb{R}^{n-1}}(\Sigma^{\prime}\cap
B_{1/4})\supset\mathbb{R}^{n-1}\cap B_{1/8}$, and hence by (6.15)
$u:\mathbb{R}^{n-1}\cap B_{1/8}\to\mathbb{R}$ is $C^{1,1/2}$ with
$|u|_{C^{1,1/2}}\leq c(n,\theta)E$. This proves our first claim.

Take $x\in\mathrm{spt}V_{I}\cap B_{1/8}$, and from our claim above we can
choose $y\in\mathrm{spt}V_{I}\cap\mathbb{R}^{n}\cap B_{1/8}$ with
$(y_{2},\ldots,y_{n})=(x_{2},\ldots,x_{n})$. Note this means that
$|x-y|=(x_{1}^{2}+x_{n+1}^{2})^{1/2}\leq 1/8$. From Step 2 we have an
$n$-plane $Q_{y}=q(P_{\theta_{y}})$ for some $q\in O(n)$ and some $\theta_{y}$
with $|q_{y}-Id|+|\theta_{y}-\theta|\leq c(n,\theta)E$ so that

$\mathrm{osc}_{Q_{y}}(V_{I},B_{2|x-y|}(y))\leq c(n,\theta)E|x-y|^{3/2}.$
(6.16)

So $x\in B_{c(n,\theta)E|x-y|}(y+Q_{y})\subset
B_{2c(n,\theta)E|x-y|}(y+P_{\theta})$, and hence ensuring
$\varepsilon(n,\theta)$ is sufficiently small we get

$2|x-y|\cos\theta>|x_{1}|>|x-y|\cos\theta/2=:R.$ (6.17)

By (6.16) and Lemma 3.15, as $\varepsilon\to 0$ we have $V\to
q_{y}(V^{(\theta_{y})})$ in $B_{2|x-y|}(y)$, hence by taking
$\varepsilon(n,\theta)$ smaller, if needed, we have the bound

$\Theta_{V}(x,R)<3/2.$ (6.18)

From (6.16), (6.17), (6.18) we can apply Allard’s theorem 2.16 in $B_{R}(x)$
to get that

$\mathrm{spt}V_{I}\cap B_{R/2}(x)=\mathrm{graph}_{Q_{y}}(w)\cap B_{R/2}(x)$

for $w:Q_{y}\cap B_{R/2}(x)\to\mathbb{R}$ a $C^{1,1/2}$ function satisfying
satisfying the scale-invariant estimates

$\displaystyle|R^{-1}w(R\cdot)|_{C^{1,1/2}}$ $\displaystyle\leq
c(n)\max\\{R^{-1}\mathrm{osc}_{Q_{y}}(V_{I},B_{R}(x)),R||H_{V,g}||_{L^{\infty}(B_{R}(x))},R|Dg|_{C^{0}(B_{R}(x))}\\}$
$\displaystyle\leq c(n,\theta)ER^{1/2}.$ (6.19)

Let now $Q_{x}$ be the tangent plane to $\mathrm{graph}_{Q_{y}}(w)$ at $x$.
Then from (6.19) we can assume

$|Q_{x}-Q_{y}|\leq c(n,\theta)ER^{1/2}$

(which implies $|Q_{x}-P_{\theta}|\leq c(n,\theta)E$), and for $0<r\leq R/4$
we have the estimate

$r^{-1}\mathrm{osc}_{Q_{x}}(V_{I},B_{r}(x))\leq c(n,\theta)Er^{1/2}.$

If $R/4<r<1/8$ we can use (6.16), (6.19) to estimate instead

$\displaystyle r^{-1}\mathrm{osc}_{Q_{x}}(V_{I},B_{r}(x))$ $\displaystyle\leq
cER^{\prime 1/2}+cr^{-1}\mathrm{osc}_{Q_{y}}(V_{I},B_{r+|x-y|}(y))$ (6.20)
$\displaystyle\leq cER^{\prime 1/2}+cEr^{-1}(r+R)^{3/2}$ (6.21)
$\displaystyle\leq c(n,\theta)Er^{1/2}.$ (6.22)

And of course (6.2) is trivial if $1/8\leq r<1/2$. This complete the proof of
Theorem 6.1. ∎

######  Theorem 6.3.

For any $\eta>0$, there are $\varepsilon(n,\eta,\theta_{0})$,
$c(n,\theta_{0})$ positive so that if $V$ satisfies Assumptions 5.1, and it
additionally satisfies

$\displaystyle
E:=\max\left\\{\mathrm{osc}_{P_{\beta(0)}}(V_{I},B_{1}),\frac{||H^{tan}_{V,g}||_{L^{\infty}(B_{1})}}{\varepsilon},\frac{|D\beta|_{C^{0}(B_{1})}}{\varepsilon},\frac{|g-g_{eucl}|_{C^{1}(B_{1})}}{\varepsilon}\right\\}\leq\varepsilon,$
(6.23) $\displaystyle|\beta(0)-\theta_{0}|\leq\varepsilon,$ (6.24)

then we can find a $q\in O(\\{x_{1}=0\\})$ with $|q-Id|\leq cE$ so that

$\displaystyle\mathrm{osc}_{q(P_{\beta(0)})}(V_{I},B_{\eta})\leq c\eta^{2}E.$
(6.25)

###### Proof.

For ease of notation write $\theta=\theta_{0}$. Suppose, towards a
contradiction, the Theorem is false. Then for any $\eta(n,\theta),c(n,\theta)$
predetermined, there are sequences $\varepsilon_{i}\to 0$, $g_{i}$,
$\beta_{i}$, and $V_{i}$ $n$-rectifiable varifolds in $(B_{1},g_{i})$
satisfying Assumption 5.1 and (6.24) (with
$V_{i},E_{i},\beta_{i},g_{i}\varepsilon_{i}$ in place of
$V,E,\beta,g,\varepsilon$), but for which (6.25) fails for all $q\in O(n)$
with $|q-I|\leq cE_{i}$.

We note that in $B_{1}$, $V_{i}\to V^{(\theta)}$ as varifolds by Lemma 3.15,
and hence by Lemma 3.12 $V_{i,I}\to[P^{-}_{\theta}]$ as varifolds and
$\mathrm{spt}V_{i,I}\to P^{-}_{\theta}$ in the local Hausdorff distance.
Define $\theta_{i}=\beta_{i}(0)$, so that $\theta_{i}\to\theta$ also.

_Step 1: Inhomogenous blow-up to graph._ For each $i$ define the map
$F_{i}:\mathbb{R}^{n+1}\to\mathbb{R}^{n+1}$ by

$F_{i}(x_{1},\ldots,x_{n},x_{n+1})=(x_{1},\ldots,x_{n},E_{i}^{-1}(x_{n+1}+x_{1}\cot\theta_{i})),$

and then define the closed sets

$\Sigma_{i}=F_{i}(\mathrm{spt}V_{i,I})\cap C^{\theta}_{1/4}(0).$

By (6.24), each $\Sigma_{i}$ is a closed subset of
$C^{\theta}_{1/4}(0)\cap\\{|x_{n+1}|\leq 1/\sin\theta_{i}\\}$, and
$0\in\Sigma_{i}$, and so after passing to a subsequence we can assume
$\Sigma_{i}\to\Sigma$ in the Hausdorff metric, for some non-empty closed
subset $0\in\Sigma\subset C^{\theta}_{1/4}(0)\cap\\{|x_{n+1}|\leq
1/\sin\theta\\}$. For $x^{\prime}\in B^{\theta}_{1/4}\cap\\{x_{1}\leq 0\\}$,
define $\textbf{u}(x^{\prime})=\\{y:(x^{\prime},y)\in\Sigma\\}$.

We first claim that $\textbf{u}(x^{\prime})\neq\emptyset$. If we had
$\textbf{u}(x^{\prime})=\emptyset$ for some $x^{\prime}=\pi_{{P_{\pi/2}}}(x)$,
$x\in P^{-}_{\theta}$, then by closedness there would be a radius $r>0$ for
which $C^{\theta}_{r}(x^{\prime})\cap\Sigma=\emptyset$. Therefore for $i\gg 1$
we would have $B_{r/2}(x)\cap\mathrm{spt}V_{i,I}=\emptyset$, which contradicts
the fact that $\mathrm{spt}V_{i,I}\to P^{-}_{\theta}$ as closed sets in
$B_{1}$. So we have $\\#\textbf{u}(x^{\prime})\geq 1$ for all $x^{\prime}\in
B^{\theta}_{1/4}\cap\\{x_{1}\leq 0\\}$.

We second claim that in fact $\textbf{u}(x^{\prime})=\\{u(x^{\prime})\\}$ for
$u:C^{\theta}_{1/4}\cap\\{x_{1}\leq 0\\}\to\mathbb{R}$ a $C^{\alpha}$ function
satisfying $|u|_{C^{\alpha}}\leq c(n,\theta)$, with $\alpha$ as in Theorem
5.3. To see this, note that by from Theorem 5.3 and our hypotheses there is a
sequence $\tau_{i}\to 0$ so that

$\mathrm{osc}_{P_{\theta_{i}}}(V_{I,i},B_{r}(x))\leq
cr^{\alpha}E_{i}\quad\forall x\in\mathrm{spt}V_{i,I}\cap
B_{1/2},\quad\forall\tau_{i}\leq r\leq 1/2.$ (6.26)

Take $x^{\prime}\neq y^{\prime}\in B^{\theta}_{1/4}\cap\\{x_{1}\leq 0\\}$, and
$x_{n+1}\in\textbf{u}(x^{\prime}),y_{n+1}\in\textbf{u}(y^{\prime})$. We aim to
show that

$|x_{n+1}-y_{n+1}|\leq c(n,\theta)|x^{\prime}-y^{\prime}|^{\alpha}.$ (6.27)

From this, and the fact $|x_{n+1}|\leq 2/\sin\theta$, our second claim will
follow.

To prove (6.27), there is no loss in assuming $|x^{\prime}-y^{\prime}|\leq
1/8$, as otherwise the inequality is trivial. Choose
$x_{i},y_{i}\in\mathrm{spt}V_{i,I}$ so that
$F_{i}(x_{i})\to(x^{\prime},x_{n+1})$ and
$F_{i}(y_{i})\to(y^{\prime},y_{n+1})$. In other words, if we write
$x_{i}=(x_{i}^{\prime},x_{i,n+1})=(x_{i,1},x_{i,2},\ldots,x_{i,n+1})$ then
$x_{i}^{\prime}\to x^{\prime}$ and
$E_{i}^{-1}(x_{i,n+1}+x_{i,1}\cot\theta_{i})\to x_{n+1}$, and the same for
$y$.

Applying (6.26) with the balls $B_{2|x^{\prime}-y^{\prime}|}(x_{i})$, then for
$i\gg 1$ we get

$|(x_{i}-y_{i})\cdot\nu_{\theta_{i}}|\leq
c(n,\theta)|x^{\prime}-y^{\prime}|^{\alpha}E_{i},$

which implies (6.27) after dividing by $E_{i}\sin\theta_{i}$ and taking
$i\to\infty$. This proves our second claim.

_Step 2: Graph is viscosity solution._ We claim that $u$ is a viscosity
solution to

$\mathcal{L}_{\theta}u=0\text{ in
}B^{\theta}_{1/4}\cap\\{x_{1}<0\\},\quad\partial_{1}u=0\text{ on
}B^{\theta}_{1/4}\cap\\{x_{1}=0\\}.$ (6.28)

Suppose $U\subset B^{\theta}_{1/4}\cap\\{x_{1}<0\\}$ is an open set, and
$\phi:U\to\mathbb{R}$ is a smooth function satisfying $\phi\geq u$ on $U$ and
$\phi=u$ at $\bar{x}^{\prime}\in U$. We can pick numbers $c_{i}\to 0$ and
points $x_{i}\to(\bar{x}^{\prime},u(\bar{x}^{\prime}))$ so that in
$U\times\mathbb{R}$, $\mathrm{graph}(\phi+c_{i})$ touches $\Sigma_{i}$ from
above at $x_{i}$. Therefore $F_{i}^{-1}(\mathrm{graph}(\phi+c_{i}))\equiv
G_{\theta_{i}}(E_{i}(\phi+c_{i}))$ touches $\mathrm{spt}V_{i,I}$ from above at
$F_{i}^{-1}(x_{i})$ in $U\times\mathbb{R}$.

By the strong maximum principle, writing $x_{i}=(x_{i}^{\prime},x_{i,n+1})$,
we conclude that

$-\varepsilon_{i}E_{i}\leq-||H^{tan}_{V_{i},g_{i}}||\leq
H_{\theta_{i}}(E_{i}(\phi+c_{i}),g_{i})|_{x_{i}}\leq
E_{i}\mathcal{L}_{\theta}(\phi)|_{x_{i}^{\prime}}+c(\phi,n,\theta)(E_{i}^{2}+\varepsilon_{i}E_{i})$

Taking $i\to\infty$ we deduce that
$\mathcal{L}_{\theta}(\phi)|_{\bar{x}^{\prime}}\geq 0$. The same argument
shows that if $\phi$ touches $u$ at $\bar{x}^{\prime}$ from below instead then
$\mathcal{L}_{\theta}(\phi)|_{\bar{x}^{\prime}}\leq 0$. We deduce that
$\mathcal{L}_{\theta}(u)=0$ in the viscosity sense.

Suppose now $U$ is relatively open in $B^{\theta}_{1/4}\cap\\{x_{1}\leq 0\\}$,
and $\phi:U\to\mathbb{R}$ is a smooth function which touches $u$ from above at
some $\bar{x}^{\prime}\in U\cap\\{x_{1}=0\\}$. For $\delta>0$ small and
$\Lambda>0$ large (to be fixed momentarily), consider the function
$\phi_{\delta}(x^{\prime})=\phi-\Lambda x_{1}^{\prime 2}-\delta
x_{1}^{\prime}$. For $\delta^{\prime}(\Lambda,\delta)$ small, on
$U\cap\\{-\delta^{\prime}\leq x_{1}^{\prime}\leq 0\\}$ we have
$\phi_{\delta}\geq\phi$ and
$\phi_{\delta}(\bar{x}^{\prime})=\phi(\bar{x}^{\prime})$, and so
$\phi_{\delta}$ touches $u$ from above at $\bar{x}^{\prime}$ also.

As before, pick $c_{i}\to 0$ and
$x_{i}\to(\bar{x}^{\prime},u(\bar{x}^{\prime}))$ so that
$F_{i}^{-1}(\mathrm{graph}(\phi_{\delta}+c_{i}))\equiv
G_{\theta_{i}}(E_{i}(\phi_{\delta}+c_{i}))$ touches $\mathrm{spt}V_{i,I}$ from
above at $y_{i}=F_{i}^{-1}(x_{i})$. Fix $\Lambda(\phi,n,\theta)$ sufficiently
large so that

$\mathcal{L}_{\theta}(\phi_{\delta})=\mathcal{L}_{\theta}\phi-2\sin^{2}\theta\Lambda<-\Lambda/2<0\text{
on }U,$

and hence for $i\gg 1$ we have

$H_{\theta_{i}}(E_{i}(\phi_{\delta}+c_{i}),g_{i})|_{x_{i}}\leq-
E_{i}\Lambda/2+c(\phi,n,\theta)(E_{i}^{2}+\varepsilon_{i}E_{i})<-\varepsilon_{i}E_{i}\leq-||H^{tan}_{V_{i},g_{i}}||.$

By the strong maximum principle we deduce that necessarily
$x_{i}\in\\{x_{1}=0\\}$ for all $i\gg 1$.

But now by Lemma 3.15 and Theorem 4.1, we must have

$\displaystyle\cos\theta_{i}-\varepsilon_{i}E_{i}$
$\displaystyle\leq\cos\beta_{i}(x_{i})$
$\displaystyle\leq\nu_{\theta_{i}}(E_{i}(\phi_{\delta}+c_{i}),g_{i})|_{x_{i}}\cdot
e_{1}$
$\displaystyle\leq\cos\theta_{i}+E_{i}(\cos^{2}\theta\sin\theta-1)\partial_{1}\phi_{\delta}|_{x_{i}^{\prime}}+c(\phi,n,\theta)(E_{i}^{2}+\varepsilon_{i}E_{i}),$

and hence taking $i\to\infty$ we get
$\partial_{1}\phi(\bar{x}^{\prime})\leq\delta$. Since $\delta$ is arbitrary we
deduce $\partial_{1}\phi(\bar{x}^{\prime})\leq 0$. A verbatim argument shows
that if $\phi$ touches from below then instead we have
$\partial_{1}\phi(\bar{x}^{\prime})\geq 0$. We deduce $\partial_{1}u=0$ along
$B_{1/4}^{\theta}\cap\\{x_{1}=0\\}$ in the viscosity sense.

_Step 3: $C^{1,\alpha}$ decay._ From Step 2, we know that $u$ solves (6.28) in
the viscosity sense, and hence $u$ solves (6.28) in the classical sense. Since
$|u|\leq c(n,\theta)$, standard Schauder estimates (e.g. after reflection
about $\\{x_{1}=0\\}$) give that

$\sup_{B^{\theta}_{1/8}\cap\\{x_{1}\leq 0\\}}\Big{(}|Du|+|D^{2}u|\Big{)}\leq
c(n,\theta),$

and therefore

$|u(x^{\prime})-Du|_{0}(x^{\prime})|\leq c(n,\theta)|x^{\prime}|^{2}\text{ on
}B^{\theta}_{1/8}\cap\\{x_{1}\leq 0\\}.$

By our Hausdorff convergence $\Sigma_{i}\to\mathrm{graph}(u)$ on
$C^{\theta}_{1/4}$, we deduce that that for any $\eta\in(0,1/8)$, we have for
all $i\gg 1$:

$\sup\\{|x_{n+1}-Du|_{0}(x^{\prime})|:(x^{\prime},x_{n+1})\in\Sigma_{i}\cap
C^{\theta}_{\eta}\\}\leq c(n,\theta)\eta^{2}.$ (6.29)

The Neumann condition $\partial_{1}u=0$ implies $Du|_{0}(e_{1})=0$, so we can
find $q_{i}\in O(\\{x_{1}=0\\})$ and linear functions
$\ell_{i}:{P_{\pi/2}}\to\mathbb{R}$ satisfying

$q_{i}(P_{\theta_{i}})=G_{\theta_{i}}(\ell_{i}),\quad|\ell_{i}-E_{i}Du|_{0}|\leq
c(n,\theta)E_{i}^{2},\quad|q_{i}-Id|\leq c(n,\theta)E_{i}.$ (6.30)

From the definition of $\Sigma_{i},F_{i}$, and from (6.29), (6.30) we deduce
that

$\mathrm{osc}_{q_{i}(P_{\theta_{i}})}(V_{I,i},B_{\eta})\leq
c\eta^{2}E_{i}+cE_{i}^{2}\leq c(n,\theta)\eta^{2}E_{i}\quad\forall i\gg 1.$

This gives (6.25), which violates our contradiction hypothesis for $i$ large.
∎

##  7\. Regularity and Main theorems

Our main regularity theorem is the following.

######  Theorem 7.1 (Allard-type regularity).

Given $\theta\in(0,\pi/2)$, there are constants $c(n,\theta)$,
$\varepsilon(n,\theta)$ so that the following holds. Let $g$ be a $C^{1}$
metric on $B_{1}$, $\beta$ a $C^{1}$ function on $B_{1}$, and $V$ a
$(\beta,S)$-capillary varifold in $(B_{1},g)$. Suppose

$\displaystyle 0\in\mathrm{spt}V_{I},\quad\Theta_{V}(0,1)\leq
3/4+\cos\theta/4,$ (7.1) $\displaystyle
E:=\max\\{\mathrm{osc}_{P_{\theta}}(V_{I},B_{1}),||H^{tan}_{V,g}||_{L^{\infty}(B_{1})},|D\beta|_{C^{0}(B_{1})},|g-g_{eucl}|_{C^{1}(B_{1})}\\}\leq\varepsilon,$
(7.2)

Then there is a $C^{1,1/2}$ function $u:B^{\theta}_{1/16}(0)\to\mathbb{R}$
satisfying

$\displaystyle\mathrm{spt}V_{I}\cap B_{1/16}=G_{\theta}(u)\cap B_{1/16},$
(7.3) $\displaystyle|u|_{C^{1,1/2}}\leq cE.$ (7.4)

###### Proof.

From Theorem 6.1, if we let $U=\pi_{{P_{\pi/2}}}(\mathrm{spt}V_{I}\cap
B_{1/8})$, then there is a function $u:U\to\mathbb{R}$ so that

$\mathrm{spt}V_{I}\cap B_{1/8}=G_{\theta}(u),$

and with the property that for every $x^{\prime},y^{\prime}\in U$, we have

$|u(x^{\prime})|\leq
cE,\quad|u(y^{\prime})-u(x^{\prime})-\ell_{x^{\prime}}(y^{\prime}-x^{\prime})|\leq
c|x^{\prime}-y^{\prime}|^{3/2}E$

for some linear map $\ell_{x^{\prime}}:{P_{\pi/2}}\to\mathbb{R}$ satisfying
$|\ell_{x^{\prime}}|\leq cE$. Here $c=c(n,\theta)$.

It will suffice to prove that $U\supset B_{1/16}^{\theta}\cap\\{x_{1}\leq
0\\}$, as it will then follow from standard arguments that $u$ is $C^{1,1/2}$
with norm bounded by $c(n,\theta)E$. Note that by Lemma 3.15, provided
$\varepsilon(n,\theta)$ is small the set $U$ is closed and $1/100$-dense in
$B_{1/32}^{\theta}\cap\\{x_{1}\leq 0\\}$. Suppose $x^{\prime}\in
B^{\theta}_{1/16}\cap\\{x_{1}\leq 0\\}\setminus U$. Choose $\rho>0$ so that
$B_{\rho}(x^{\prime})\cap U=\emptyset$ but there is $y^{\prime}\in
U\cap\partial B_{\rho}(x^{\prime})$.

Choose $y\in\mathrm{spt}V_{I}\cap B_{1/32}$ with
$\pi_{{P_{\pi/2}}}(y)=y^{\prime}$. By Lemma 3.12 (if $y_{1}=0$) or by Theorem
2.15 and (3.9) (if $y_{1}<0$), the constancy theorem, and the decay estimate
(6.2), we have that as $r\to 0$ the rescaled supports

$\mathrm{spt}(V_{y,r,I})\cap B_{1}\equiv(\mathrm{spt}V_{I}-y)/r\cap B_{1}\to
Q_{y}\cap\\{x_{1}\leq 0\\}$

in the Hausdorff distance as $r\to 0$. Therefore
$\pi_{{P_{\pi/2}}}((\mathrm{spt}V_{I}-y)/r)\to U_{y}$ where
$U_{y}={P_{\pi/2}}\cap\\{x_{1}<0\\}$ if $y_{1}=0$ and $U_{y}={P_{\pi/2}}$ if
$y_{1}<0$. But on the other hand, our choice of $\rho$ would necessitate that
$U_{y}\subset\\{z\in{P_{\pi/2}}:z\cdot(x-y)\leq 0\\}$, which is a
contradiction. ∎

One corollary of the regularity theorem and the classification of cones, which
was not highlighted in the Introduction, is the following cone
rigidity/density gap. Compare to Lemma 4.2.

######  Corollary 7.2.

Given $\theta\in(0,\pi/2)$, $\Lambda>0$, there is an
$\varepsilon(n,\theta,\Lambda)$ so that the following holds. Let $V$ be a non-
zero, conical, stationary $(\theta,S)$-capillary varifold in
$\mathbb{R}^{n+1}$ satisfying

$\displaystyle\Theta_{V}(0,1)\leq(1+\cos\theta)/2+\varepsilon,\quad\Lambda<\mu_{V}(B_{1}\cap\mathbb{R}^{n})<\cos\theta\omega_{n}.$
(7.5)

Then up to rotation $V=V^{(\theta)}$.

###### Proof.

Proof by contradiction: suppose there are sequences $\varepsilon_{i}\to 0$ and
$V_{i}$ satisfying (7.5) so that (up to rotation) $V_{i}\neq V^{(\theta)}$.
Since every $\mu_{V_{i}}\neq\cos\theta[\mathbb{R}^{n}]$ and $\neq 0$, from
Theorem 8.1 we have that
$\mathrm{spt}\sigma_{V_{i}}\equiv\mathrm{spt}V_{I,i}\cap\mathbb{R}^{n}\neq 0$.
By Lemma 3.12, up to a subsequence, the $V_{i}\to V$ for some conical,
stationary $(\theta,S)$-capillary varifold in $\mathbb{R}^{n+1}$ satisfying

$\Theta_{V}(0,1)\leq(1+\cos\theta)/2,\quad V_{I}\neq
0,\quad\mu_{V}(\mathbb{R}^{n})>0.$

If $\mu_{V}\llcorner\mathbb{R}^{n}=\cos\theta[\mathbb{R}^{n}]$, then $V_{I}$
is a non-zero stationary free-boundary cone with density
$\theta_{V_{I}}(0)\leq(1-\cos\theta)/2<1/2$. However, by standard monotonicity
we must have $\theta_{V_{I}}(0)\geq 1/2$, which is a contradiction. Therefore
we must have $\mu_{V}(\mathbb{R}^{n}\cap B_{1})<\cos\theta\omega_{n}$.

We can apply Lemma 4.2 to get $V=V^{(\theta)}$, and hence by Theorem 1.15 we
have for $i\gg 1$ every $V_{I,i}$ is a $C^{1}$ surface meeting
$\mathbb{R}^{n}$ at angle $\theta$. Since each $V_{i}$ is conical we must have
$V_{i}=q_{\sharp}V^{(\theta)}$, for some rotation $q$. ∎

We now prove the theorems and corollaries outlaid in the Introduction.

###### Proof of Theorem 1.6.

This is proven in Theorem 3.1 and Theorem 3.16. ∎

###### Proof of Theorem 1.8.

The proof is the same as in the capillary case, simply replace $H_{\beta,g}$
with $H^{tan}_{V}|_{x_{1}=0}$. ∎

###### Proof of Theorem 1.11.

By Lemma 3.15, as $\delta\to 0$ then $V\to V^{(\theta)}$ in $B_{1}$ and
$\beta\to\theta$ in $C^{1}(B_{1})$. If $\theta>\pi/2$, we can replace $V$ with
$V^{\prime}=V-\cos\beta[\mathbb{R}^{n}]_{g}$, and then $V^{\prime}$ is a
$(\pi-\beta,\mathbb{R}^{n}\setminus S)$-capillary varifold in $(B_{1},g)$ with
the property that $V^{\prime}\to q_{\sharp}V^{(\pi-\theta)}$ in $B_{1}$ as
$\delta\to 0$, where $q$ is reflection about the line $\\{x_{n+1}=0\\}$.
Therefore, taking $\delta(n,\theta)$ sufficiently small there is no loss in
assuming $\theta<\pi/2$, and then Theorem 1.11 is a direct consequence of
Theorem 7.1 (for the regularity) and Theorem 3.1 (for the closeness of
$\mathrm{spt}V_{I}$ to $0$). ∎

###### Proof of Theorem 1.15.

Since by our assumption $V\to V^{(\theta)}$ and $\cos\beta\to\theta$ as
$\delta\to 0$, we have $\Theta_{V}(0,1)+(\cos\beta(0))_{-}\leq
3/4+|\cos\theta|/4<1$ for $\delta(n,\theta)$ small. Lemma 3.12 then implies
$\mathrm{spt}V_{I}\to P_{\theta}^{-}$ as $\delta\to 0$, and so we can apply
Theorem 1.11 (in a slightly smaller ball) for $\delta(n,\theta)$ small. ∎

###### Proof of Corollary 1.17.

Suppose $\beta(x)\in(0,\pi)\setminus\\{\pi/2\\}$, and some tangent cone of $V$
at $x\in\mathbb{R}^{n}$ is $V^{(\beta(x))}$, i.e. if one changes coordinates
so that $g(x)=g_{eucl}$, then there is a sequence $r_{i}\to 0$ so that
$V_{i}:=(\eta_{x,r_{i}})_{\sharp}V\to V^{(\beta(x))}$, where we interpret
$V_{i}$ as $(\beta_{i},S_{i})$-capillary varifolds in the metric
$g_{i}(x):=g(rx)$, with $\beta_{i}(x)=\beta(rx)$.

Without loss of generality, let us simply suppose $x=0$ and $g(0)=g_{eucl}$.
For any $\varepsilon>0$ small and $i\gg 1$, we have

$\Theta_{V_{i}}(0,1)+(\cos\beta(0))_{-}\leq\Theta_{V}(0,1)+(\cos\beta(0))_{-}+\varepsilon=(1+|\cos\beta(0)|)/2+\varepsilon<1.$

Therefore we can use Lemma 3.12 to get $\mathrm{spt}V_{I,i}\to P_{\theta}^{-}$
in the local Hausdorff distance, and hence for any $\varepsilon>0$ and $i\gg
1$ we have

$0\in\mathrm{spt}V_{I,i},\quad\mathrm{osc}_{P_{\beta(0)}}(V_{I,i},B_{1})\leq\varepsilon.$

And of course by scaling we have

$\max\\{||H^{tan}_{V_{i},g_{i}}||_{L^{\infty}(B_{1})},|g_{i}-g_{eucl}|_{C^{1}(B_{1})},|D\beta_{i}|_{C^{0}(B_{1})}\\}\to
0$

Now apply Theorem 1.11. ∎

###### Proof of Theorem 1.19.

For the first assertion it suffices to work in a small neighborhood of some
point
$x\in\mathrm{spt}V_{I}\cap\mathbb{R}^{n}\cap\\{\Theta_{V,g}(x)+(\cos\beta(x))_{-}<1\\}$.
By upper-semi-continuity of $\Theta_{V,g}$ and monotonicity (2.62), we can
assume without loss of generality that $x=0$, and that

$\Theta_{V}(0,1)+(\cos\beta(0))_{-}<1.$

Openness of the regular set follows from the definition, so we will focus on
proving denseness in a neighborhood of $0$.

By Theorem 3.16, there is a radius $\gamma>0$ so that for
$\mathcal{H}^{n-1}$-a.e. $x\in
B_{\gamma}\cap\mathbb{R}^{n}\cap\\{\beta\neq\pi/2\\}\cap\partial^{*}S$, every
tangent cone to $V$ at $x$ is (some rotation of) $V^{(\beta(x))}$, which
implies by Corollary 1.17 that every such $x$ is a regular point.

On the other hand, Theorem 3.16 says that for $\mathcal{H}^{n-1}$-a.e.
$x\in\mathrm{spt}V_{I}\cap\mathbb{R}^{n}\cap
B_{\gamma}\setminus(\\{\beta\neq\pi/2\\}\cap\partial^{*}S)$ we have
$\eta_{V_{I},g}(x)\perp_{g}\mathbb{R}^{n}$, and every tangent cone to $V_{I}$
at $x$ is some rotation of $[P_{\pi/2}]$. Therefore $V_{I}$ is a free-boundary
varifold in
$B_{\gamma}\setminus\overline{\\{\beta\neq\pi/2\\}\cap\partial^{*}S}$ with
bounded mean curvature, whose tangent at $\mathcal{H}^{n-1}$-a.e. boundary
point is a free-boundary half-plane with multiplicity-one. The Allard-type
regularity theorem of [13] implies $V_{I}$ is regular at
$\mathcal{H}^{n-1}$-a.e. $x\in\mathrm{spt}V_{I}\cap\mathbb{R}^{n}\cap
B_{\gamma}\setminus\overline{\\{\beta\neq\pi/2\\}\cap\partial^{*}S}$. This
proves denseness near $0$.

We prove the second assertion of the Theorem. After replacing $S$ with
$\mathbb{R}^{n}\setminus S$ and $\beta$ with $\pi-\beta$ as in Remark 1.3, we
can without loss of generality assume $\cos\beta(0)>0$ (since $V_{I}$ remains
unchanged). After shrinking our ball we can therefore work under the
additional assumption that

$\Theta_{V,g}(x)<1/2+\cos\beta(x)\quad\text{and}\quad\cos\beta(x)>0\quad\forall
x\in B_{1}\cap\mathbb{R}^{n}.$

Since $\Theta_{V,g}(x)=(1+\cos\beta(x))/2$ for $\mathcal{H}^{n-1}$-a.e.
$x\in\partial^{*}S\cap B_{\gamma}$, by upper-semi-continuity and our
assumptions we have

$1/2<(1+\cos\beta(x))/2\leq\Theta_{V,g}(x)<1/2+\cos\beta(x)\quad\forall x\in
B_{\gamma}\cap\overline{\partial^{*}S}.$

But $\Theta_{V,g}(x)\in\\{1/2,1/2+\cos\beta(x)\\}$ for
$\mathcal{H}^{n-1}$-a.e. $x\in\mathrm{spt}V_{I}\cap\mathbb{R}^{n}\cap
B_{\gamma}\setminus\partial^{*}S$, and so we must have
$\mathcal{H}^{n-1}(B_{\gamma}\cap\overline{\partial^{*}S}\setminus\partial^{*}S)=0$.
In $B_{\gamma}\setminus\overline{\partial^{*}S}$ the interior varifold $V_{I}$
is a free-boundary varifold, and so by the same reasoning as before using [13]
we have $\mathcal{H}^{n-1}$-a.e. $x\in\mathrm{spt}V_{I}\cap\mathbb{R}^{n}\cap
B_{\gamma}\setminus\overline{\partial^{*}S}$ is a regular point. ∎

##  8\. Appendix: free-boundary first variation control

Recall that a signed $n$-rectifiable varifold $V$ in
$(B_{1}\subset\mathbb{R}^{n+1},g)$ is said to have free-boundary in
$\\{x_{1}=0\\}$ if there is a vector field $H^{tan}_{V,g}\in
L^{1}_{loc}(B_{1},\mathbb{R}^{n+1};|\mu_{V}|)$ so that $H_{V,g}^{tan}$ is
tangential to $\\{x_{1}=0\\}$ at $|\mu_{V}|$-a.e. point in $\\{x_{1}=0\\}$ and

$\delta_{g}V(X)=-\int g(H_{V,g}^{tan},X)d\mu_{V}$ (8.1)

for all $X\in C^{1}_{c}(B_{1},\mathbb{R}^{n+1})$ which are tangential to
$\\{x_{1}=0\\}$.

######  Theorem 8.1.

Let $g$ be a $C^{1}$ metric on $B_{1}$, and let $V$ be an $n$-rectifiable
signed varifold in $(B_{1},g)$ supported in $\\{x_{1}\leq 0\\}$ with $g$-free-
boundary in $\\{x_{1}=0\\}$ and $g$-mean curvature $H^{tan}_{V,g}$, and assume
that $\mu_{V_{I}}\equiv\mu_{V}\llcorner\\{x_{1}<0\\}$ is a positive measure.
Then $V$ has locally-finite $g$-first-variation in $B_{1}$, and there exists a
Radon measure $\sigma_{V,g}\perp|\mu_{V}|$ on $B_{1}$, supported in
$\\{x_{1}=0\\}$, so that

$\delta_{g}V(Y)=-\int
g(H_{V,g}^{tan},Y)d\mu_{V}-\int_{\\{x_{1}=0\\}}g(H_{\mathbb{R}^{n},g},Y)d\mu_{V}+\int
g(\nu_{g},Y)d\sigma_{V,g}$ (8.2)

where $H_{\mathbb{R}^{n},g}$ is the mean curvature vector of
$\mathbb{R}^{n}\subset(\mathbb{R}^{n+1},g)$ and $\nu_{g}$ is defined in (2.4).

Furthermore, for any $\gamma<1$ we have the boundary measure mass bound

$\sigma_{V,g}(B_{\gamma})\leq
c||H^{tan}_{V,g}||_{L^{1}(\mu_{V_{I}})}+c(|Dg|_{C^{0}(B_{1})}+1)\mu_{V_{I}}(B_{1})$
(8.3)

where $c=(n,\gamma,|g|_{C^{0}(B_{1})})$.

######  Remark 8.2.

Theorem 8.1 says that if $V$ has bounded tangential mean curvature
$H^{tan}_{V,g}$, then $V$ has bounded total mean curvature
$H_{V,g}=H_{V,g}^{tan}+H_{\mathbb{R}^{n},g}$. Since $H_{V,g}$ coincides with
$H^{tan}_{V,g}$ for tangential fields and
$|H_{V,g}|\leq|H^{tan}_{V,g}|+c(n)|Dg|$ (and we will always working in a
$C^{1}$ metric), in this paper we simply say $V$ has bounded mean curvature to
indicate either tangential or total mean curvature is bounded.

######  Remark 8.3.

If $V_{I}\llcorner B_{r}(x)=0$ for some $B_{r}(x)\subset B_{1}$, then
$\sigma_{V,g}\llcorner B_{r}(x)=0$ also. To see this, take any $X\in
C^{1}_{c}(B_{r}(x))$ and write

$X=(X-g(X,\nu_{g})\nu_{g})+g(X,\nu_{g})\nu_{g}=:X^{\prime\prime}+X^{\prime},$

where $\nu_{g}=g^{1j}e_{j}/\sqrt{g^{11}}$. Then $X^{\prime\prime}$ is
tangential to $\mathbb{R}^{n}$, and since $V\equiv V_{B}$ in $B_{r}(x)$ we can
argue as in the proof of Theorem 8.1:

$\displaystyle\delta_{g}V(X)$
$\displaystyle=\delta_{g}V(X^{\prime\prime})+\delta_{g}V(X^{\prime})$ (8.4)
$\displaystyle=-\int
g(H_{V,g}^{tan},X^{\prime\prime})+g(X,\nu_{g})\mathrm{div}_{\mathbb{R}^{n},g}(\nu_{g})d\mu_{V_{B}}$
(8.5) $\displaystyle=-\int
g(H_{V,g}^{tan},X^{\prime\prime})+g(H_{\mathbb{R}^{n},g},X^{\prime})d\mu_{V_{B}}$
(8.6) $\displaystyle=-\int g(H_{V,g}^{tan}+H_{\mathbb{R}^{n},g},X)d\mu_{V}.$
(8.7)

###### Proof of Theorem 8.1.

Write $|g|_{C^{0}}=\Gamma$. Define the vector field

$e_{g}=\mathrm{grad}_{g}(x_{1})\equiv\sum_{j=1}^{n+1}g^{1j}e_{j},$

and note that $e_{g}$ is $g$-orthogonal to $\\{x_{1}=0\\}$.

Take any $h\in C^{1}_{c}(B_{1})$, any $\tau\in(0,\infty)$, let
$\phi:\mathbb{R}\to\mathbb{R}$ be a cutoff function which $\equiv 0$ in a
neighborhood of $0$ and $\equiv 1$ on $[1,\infty)$, and then define the vector
field $X:B_{1}\to\mathbb{R}^{n+1}$ by

$X=\phi(-x_{1}/\tau)h(x)e_{g}(x).$

Trivially $X$ is compactly supported in $B_{1}\cap\\{x_{1}\neq 0\\}$, and so
by the free-boundary condition (8.1) we have

$\delta_{g}V(X)=-\int g(H_{V,g}^{tan},X)d\mu_{V}=-\int\phi hH_{V,g}^{tan}\cdot
e_{1}d\mu_{V}$

We compute

$\displaystyle-\int\phi\left(\frac{-x_{1}}{\tau}\right)hH_{V,g}^{tan}\cdot
e_{1}d\mu_{V}$ $\displaystyle=\int\mathrm{div}_{V,g}(X)d\mu_{V}$
$\displaystyle=\int\left(-\frac{\phi^{\prime}}{\tau}hg(\pi_{V,g}(\mathrm{grad}_{g}(x_{1})),e_{g})+\phi\mathrm{div}_{V,g}(he_{g})\right)d\mu_{V}$
$\displaystyle=\int\left(-\frac{\phi^{\prime}}{\tau}h|\pi_{V,g}(e_{g})|^{2}_{g}+\phi\mathrm{div}_{V,g}(he_{g})\right)d\mu_{V}.$

By the dominated convergence theorem we can assume without loss of generality
that $\phi(s)=\max\\{\min\\{s,1\\},0\\}$, and we therefore obtain

$\frac{1}{\tau}\int_{\\{-\tau<x_{1}<0\\}}h|\pi_{V,g}(e_{g})|^{2}_{g}d\mu_{V}=\int_{\\{x_{1}<0\\}}\left(\phi
hH_{V,g}^{tan}\cdot e_{1}+\phi\mathrm{div}_{V,g}(he_{g})\right)d\mu_{V}.$

Once again by dominated convergence, taking $\tau\to 0$ we deduce

$\displaystyle\lim_{\tau\to
0}\frac{1}{\tau}\int_{\\{-\tau<x_{1}<0\\}}h|\pi_{V,g}(e_{g})|^{2}_{g}d\mu_{V}$
(8.8) $\displaystyle\qquad=\int_{\\{x_{1}<0\\}}\left(hH_{V,g}^{tan}\cdot
e_{1}+\mathrm{div}_{V,g}(he_{g})\right)d\mu_{V}$ (8.9)
$\displaystyle\qquad\leq
c(n)|g|_{C^{0}}|h|_{C^{1}}\Big{(}||H_{V,g}^{tan}||_{L^{1}(\mu_{V_{I}})}+(|Dg|_{C^{0}(B_{1})}+1)\mu_{V_{I}}(B_{1})\Big{)}$
(8.10)

where $\mu_{V_{I}}=\mu_{V}\llcorner\\{x_{1}<0\\}$.

Now take $\gamma<1$, and fix $\Phi$ any cutoff function which is $\equiv 1$ in
$B_{\gamma}$ and supported in $B_{1}$. We claim that

$F(h):=\lim_{\tau\to
0}\frac{1}{\tau}\int_{\\{-\tau<x_{1}<0\\}}h|\pi_{V,g}(e_{g})|^{2}d\mu_{V}$
(8.11)

is a Radon measure on $B_{\gamma}$, i.e. $F(h)=\int hdF$. To see this, simply
observe that

$\displaystyle F(h)$ $\displaystyle\leq|h|_{C^{0}(B_{\gamma})}F(\Phi)$ (8.12)
$\displaystyle\leq|h|_{C^{0}(B_{\gamma})}c(n,\gamma,\Gamma)(||H_{V,g}^{tan}||_{L^{1}(\mu_{V_{I}})}+(|Dg|_{C^{0}(B_{1})}+1)\mu_{V_{I}}(B_{1})),$
(8.13)

which implies that $F$ extends to a continuous linear function on
$C^{0}(B_{\gamma})$. From the definition (8.11) (and since
$\mathrm{spt}V\subset\\{x_{1}\leq 0\\}$) it’s clear that
$\mathrm{spt}F\subset\\{x_{1}=0\\}$, but also observe that $F$ only depends on
$V\llcorner\\{x_{1}<0\\}$, and so in fact $F\perp|\mu_{V}|$.

Now let $Y$ be any $C^{1}_{c}$ vector field in $B_{\gamma}$, and define

$Y^{\prime}:=\frac{g(Y,e_{g})}{|e_{g}|^{2}_{g}}e_{g}\equiv\frac{Y^{1}}{g^{11}}g^{1j}e_{j}.$

Then from (8.9) we have

$\displaystyle\int_{\\{x_{1}<0\\}}\left(\mathrm{div}_{V,g}(Y^{\prime})+g(H_{V,g}^{tan},Y^{\prime})\right)d\mu_{V}$
$\displaystyle=F(g(Y,e_{g})/|e_{g}|^{2}_{g})$ $\displaystyle\leq
c(n,\gamma,\Gamma)|Y|_{C^{0}(B_{\gamma})}(||H_{V,g}^{tan}||_{L^{1}(\mu_{V_{I}})}+(|Dg|_{C^{0}(B_{1})}+1)\mu_{V_{I}}(B_{1})).$
(8.14)

Since $V$ is $n$-rectifiable we have $T_{x}V=T_{x}\mathbb{R}^{n}$ for
$\mu_{V}$-a.e. $x_{1}=0$. Therefore, because $e_{g}/|e_{g}|_{g}$ is the
$g$-unit normal to $\\{x_{1}=0\\}$, we can compute

$\displaystyle\int_{\\{x_{1}=0\\}}\mathrm{div}_{V,g}(Y^{\prime})d\mu_{V}$
$\displaystyle=\int_{\\{x_{1}=0\\}}\frac{g(Y,e_{g})}{|e_{g}|_{g}}\mathrm{div}_{\mathbb{R}^{n},g}\left(\frac{e_{g}}{|e_{g}|_{g}}\right)d\mu_{V}$
$\displaystyle=\int_{\\{x_{1}=0\\}}-g(H_{\mathbb{R}^{n},g},Y^{\prime})d\mu_{V},$
(8.15)

where $H_{\mathbb{R}^{n},g}$ is the mean curvature vector of $\mathbb{R}^{n}$
in $(\mathbb{R}^{n+1},g)$.

Noting that both $Y^{\prime\prime}:=Y-Y^{\prime}$ and $H^{tan}_{V,g}$ are
tangential to $\\{x_{1}=0\\}$, while $Y^{\prime}$ is $g$-normal to
$\\{x_{1}=0\\}$, we can combine (8.14), (8.15) and the free-boundary condition
(8.1) to obtain

$\displaystyle\int\mathrm{div}_{V,g}(Y)d\mu_{V}$ (8.16)
$\displaystyle\quad=\int\mathrm{div}_{V,g}(Y^{\prime\prime})d\mu_{V}+\int_{\\{x_{1}=0\\}}\mathrm{div}_{V,g}(Y^{\prime})d\mu_{V}+\int_{\\{x_{1}<0\\}}\mathrm{div}_{V,g}(Y^{\prime})d\mu_{V}$
(8.17) $\displaystyle\quad=-\int
g(H_{V,g}^{tan},Y)d\mu_{V}-\int_{\\{x_{1}=0\\}}g(H_{\mathbb{R}^{n},g},Y)d\mu_{V}+F(g(Y,e_{g})/|e_{g}|^{2}_{g})$
(8.18) $\displaystyle\quad\leq
c(n,\gamma,\Gamma)|Y|_{C^{0}(B_{\gamma})}(||H_{V,g}^{tan}||_{L^{1}(B_{1})}+(|Dg|_{C^{0}(B_{1})}+1)|\mu_{V}|(B_{1})).$
(8.19)

We deduce that $V$ has locally-bounded first variation in $B_{1}$, and hence
by standard Radon-Nikodyn we can write

$\int\mathrm{div}_{V,g}(Y)d\mu_{V}=-\int g(H,Y)d\mu_{V}+\int
g(\eta,Y)d\sigma_{V}$

for some $\sigma_{V}\perp|\mu_{V}|$, and some vector field $H\in
L^{1}_{loc}(|\mu_{V}|)$, and some $g$-unit, $\sigma_{V}$-measurable vector
field $\eta$. From the free-boundary condition (8.1) it’s clear that:
$H=H^{tan}_{V,g}$ $|\mu_{V}|$-a.e. $x_{1}<0$; $g(H,X)=g(H^{tan}_{V,g},X)$ for
$|\mu_{V}|$-a.e. $x_{1}=0$ and any $X\in\mathbb{R}^{n}$; $\sigma_{V}$ must be
supported in $\mathbb{R}^{n}$; and $\eta=\pm\nu_{g}$ $\sigma_{V}$-a.e.

To prove (8.2) it remains to show that
$g(H,e_{g})=g(H_{\mathbb{R}^{n},g},e_{g})$ on $\\{x_{1}=0\\}$ and
$\eta=+\nu_{g}$. Plugging $Y=he_{g}$ into the first variation formula we have

$\displaystyle-\int g(H,e_{g})hd\mu_{V}+\int g(\eta,e_{g})hd\sigma_{V}$ (8.20)
$\displaystyle=-\int_{\\{x_{1}<0\\}}g(H^{tan}_{V,g},e_{g})hd\mu_{V}-\int_{\\{x_{1}=0\\}}g(H_{\mathbb{R}^{n},g},e_{g})hd\mu_{V}+F(h)$
(8.21)

for any $h\in C^{0}_{c}(B_{1})$. Therefore since $H=H^{tan}_{V,g}$
$\mu_{V}$-a.e. $x_{1}<0$ and since $F\perp|\mu_{V}|$, we deduce

$\int_{\\{x_{1}=0\\}}g(H-H_{\mathbb{R}^{n},g},e_{g})hd\mu_{V}=0,\quad\int
g(\eta,e_{g})hd\sigma_{V}=F(h)\geq 0$ (8.22)

for all $h\geq 0$, which implies our claim.

The mass bound (8.3) follows from (8.12) and (8.22): for any $h\in
C^{0}_{c}(B_{\gamma})$ non-negative we have

$\displaystyle\sigma_{V}(h)$ $\displaystyle\leq c(n,\Gamma)\int
g(\nu_{g},e_{g})hd\sigma_{V}$ (8.23) $\displaystyle=c(n,\Gamma)F(h)$ (8.24)
$\displaystyle\leq
c(n,\gamma,\Gamma)|h|_{C^{0}}(||H^{tan}_{V,g}||_{L^{1}(\mu_{V_{I}})}+(|Dg|_{C^{0}(B_{1})}+1)\mu_{V_{I}}(B_{1})).\qed$
(8.25)

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