A global bound for the singular set of area-minimizing hypersurfaces

TL;DR

This paper establishes a universal bound on the size of the singular set of area-minimizing hypersurfaces based on boundary geometry, advancing understanding of their regularity properties.

Contribution

It provides the first a priori measure bound for the singular set of area-minimizing hypersurfaces in relation to boundary geometry.

Findings

Bound on (n-7)-dimensional measure of singular set

Relation between boundary geometry and singular set size

Improved regularity estimates for hypersurfaces

Abstract

We give an a priori bound on the (n-7)-dimensional measure of the singular set for an area-minimizing n-dimensional hypersurface, in terms of the geometry of its boundary.

Equations

$${\mathcal{H}}^{n-7}(\mathrm{sing}T)\le c(n,\alpha )\frac{||T||({\mathbb{R}}^{n+1})}{r_{1,\alpha }(S{)}^7}$$

$${\mathcal{H}}^{n-7}(\mathrm{sing}T)\le c^{\prime }(n,\alpha )\frac{{\mathcal{H}}^{n-1}(S{)}^{\frac{n}{n-1}}}{r_{1,\alpha }(S{)}^7}.$$

$${\mathcal{H}}^{n-7}(\mathrm{sing}T)\le c(n)\frac{||T||(\Omega )}{r_{1,1}(\partial \Omega {)}^7}.$$

$${\mathcal{H}}^{n-7}(\mathrm{sing}T\cap U\setminus B_{ϵ}(\partial U))\le c(n){ϵ}^{-7}||T||(U\setminus B_{ϵ/2}(\partial U)).$$

$$||T||(B_r(x))\le ||T+S||(B_r(x))+c_0{r}^{n+2\alpha },$$

$${\mathcal{H}}^{n-7}(\mathrm{sing}T)\le c(n,\alpha )\max \{c_0^{\frac{7}{2\alpha }},r_{1,\alpha }(\partial \Omega {)}^{-7},r_{1,\alpha }(S{)}^{-7}\}||T||({\mathbb{R}}^{n+1}).$$

$$\underset{y\in S}{\inf }\frac{|x-y|}{r_{1,\alpha }(S,y)}\ge {ϵ}_1(n,\alpha ).$$

$$\mathrm{dist}(x_i,S_i)\le r_{1,\alpha }(S,y_i)/i.$$

$$h(t,x)=\left\{\begin{array}[]{l l}x+tu_{i}(x)&t\geq 0\\ x-t\sqrt{1-|x|^{2}}e_{1}&t\leq 0\end{array}\right.,$$

$$T_i-R_i=\sum _j\partial [E_{i,j}],||T_i-R_i||=\sum _i||\partial [E_{i,j}]||.$$

$$||T_i||=||\partial [E_{i,j_i}]+R_i||+\sum _{j\ne j_i}||\partial [E_{i,j}]||,$$

$$||\partial [E_{i,j}]||(B_r)\le c(n)r^n\forall r<1/4.$$

$$T_i^{\prime }:=\partial [({\eta }_{{\lambda }_i}{)}_{♯}{E}_{i,j_1}]+({\eta }_{{\lambda }_i}{)}_{♯}{R}_i.$$

$${\mathcal{H}}^{n-7}(\mathrm{sing}T\cap B_{1/2})\le c(n)||T||(B_1).$$

$${\mathcal{H}}^{n-7}(\mathrm{sing}\partial [E_i]\cap B_{1/2})\le c(n)\le c(n)||\partial [E_i]||(B_1).$$

$${\mathcal{H}}^{n-7}(\mathrm{sing}T\cap B_{1/2})$$

$$\le c(n)\sum _i||\partial [E_i]||(B_1)$$

$$=c(n)||T||(B_1).\text{qed}$$

$${\mathcal{H}}^{n-7}(\mathrm{sing}T\cap B_{ϵ/2}(x_j))\le \frac{c(n)}{{ϵ}^7}||T||(B_{ϵ}(x_j)).$$

$${\mathcal{H}}^{n-7}(\mathrm{sing}T)$$

$$\le \sum _j{\mathcal{H}}^{n-7}(\mathrm{sing}T\cap B_{ϵ/2}(x_j))$$

$$\le c(n,\alpha )\sum _j||T||(B_{ϵ}(x_j))$$

$$\le c(n,\alpha )||T||({\mathbb{R}}^{n+1})$$

$$\le c(n,\alpha ){\mathcal{H}}^{n-1}(S{)}^{n/(n-1)}.\text{qed}$$

$${\mathcal{H}}^{n-7}(\mathrm{sing}T\cap B_{ϵr/2}(x))\le c(n)||T||(B_{ϵr}(x))$$

$$T^{\prime }=\sum _i\partial [E_i],||T^{\prime }||=\sum _i||\partial [E_i]||,$$

$$T=\sum _i\partial [E_i]└\Omega ,||T||=\sum _i||\partial [E_i]└\Omega ||.$$

$$||\partial [E_i]└\Omega ||(B_r(x))\le c(n)r^n\forall B_r(x)\subset \subset B_2.$$

$$||T^{\prime }||(B_r(x))\le ||T^{\prime }+S||(B_r(x))+c(n)\Gamma r||T^{\prime }||(B_r(x)).$$

$${\tilde{\theta }}_T(x,r)=r^{-n}||T||(B_r(x))+r^{-n}||T||(\{y:|\sigma (y)-x|<r\}).$$

$${\int }_{B_r(x)\setminus B_s(x)}|y-x{|}^{-n-2}\left(|(y-x{)}^{\perp }{|}^2+|i_y(\tilde{y}-x{)}^{\perp }{|}^2\right)d||T||(y)$$

$$\le {\tilde{\theta }}_T(x,r)-{\tilde{\theta }}_T(x,s)+c(n)\Gamma r{\tilde{\theta }}_T(n,r).$$

$$\displaystyle\Big{|}||T_{i}||(\{y:|\sigma_{i}(y)-x|<r\})-||T_{i}||(\sigma(B_{r}(x))\Big{|}$$

$$\le c||T_i||(B_{(1+{\kappa }_i)r}(x)\setminus B_{(1-{\kappa }_i)r}(x)),$$

Full text

Area-minimizing surfaces in general will not be smooth, and a basic question in minimal surface theory is to understand the size and nature of the singular region. The cumulative works of many (Federer, De Giorgi, Allard, Simons, to name only a few) prove that for absolutely-area-minimizing $n$-dimensional hypersurfaces in $\mathbb{R}^{n+1}$ (“codimension-one area-minimizing integral currents”), the interior singular set is at most $(n-7)$-dimensional. This dimension bound is sharp, and is directly tied to the existence of low-dimensional, non-flat minimizing cones.

[HS79] proved that for such codimension-one area-minimizers, if the boundary is known to be $C^{1,\alpha}$ and multiplicity-one, then in fact no singularities lie within a neighborhood of the boundary. Combined with interior regularity, this theorem gives a very nice structure of these minimizing hypersurfaces.

Recently [NV17], [NV15] quantified the interior partial regularity, by demonstrating effective local (interior) bounds on the $\mathcal{H}^{n-7}$ measure of the singular set. Their methods also prove $(n-7)$-rectifiability of the singular set, which was originally established through an entirely different approach by [Sim95].

In this short note, we obtain obtain a global, effective a priori estimate on the singular set of an area-minimzing hypersurface in terms of the boundary geometry. Our results are loosely analogous to the a priori bounds of [AL88] (see also the recent works [MMS18b], [MMS18a]).

We work in $\mathbb{R}^{n+1}$, for $n\geq 7$. Let us write $\mathcal{I}_{n}(U)$ for the space of integral $n$-currents acting on forms supported in the open set $U$. Given an $n$-dimensional, oriented manifold $E$, write $[E]$ for the current induced by integration. Let $\eta_{\lambda}(x)=\lambda x$, and $\tau_{y}(x)=x+y$.

If $T\in\mathcal{I}_{n}(U)$, we say $T$ is area-minimizing if $||T||(W)\leq||T+S||(W)$ for every open $W\subset\subset U$, and every $S\in\mathcal{I}_{n}(U)$ satisfying $\partial S=0$, $\mathrm{spt}S\subset W$. The regular set $\mathrm{reg}T$ is the (open) set of points where $\mathrm{spt}T$ is locally the union of embedded $C^{1,\alpha}$ manifolds. The singular set is $\mathrm{sing}T=\mathrm{spt}T\setminus\mathrm{reg}T$. Write $||T||$ for the mass measure of $T$.

Given an $k$-manifold $S$, and $x\in S$, let $r_{1,\alpha}(S,x)$ be the largest radius $r$, so that $(S-x)/r$ is the graph of a $C^{1,\alpha}$ function $u$, with $|u|_{1,\alpha}\leq 1$. Define $r_{1,\alpha}(S)=\inf_{x\in S}r_{1,\alpha}(S,x)$.

Our main Theorem is the following.

[TABLE]

In particular, we have

[TABLE]

I believe Theorem 0.1 should hold for more general $S$, but there are subtleties even in the idealized case when $S$ is a line. See the discussion below.

We also have a version of Theorem 0.1 in the case when $T$ has free-boundary. Given open sets $U$, $\Omega$, we say $T\in\mathcal{I}_{n}(U)$ is area-minimizing with free-boundary in $\Omega$ if: $\mathrm{spt}T\subset\overline{\Omega}$, and $||T||(W)\leq||S+T||(W)$ for all $W\subset\subset U$, and every $S\in\mathcal{I}_{n}(U)$ satisfying $\mathrm{spt}S\subset\overline{\Omega}\cap W$ and $\mathrm{spt}(\partial S)\subset\partial\Omega$. [Gru87] proved boundary singularities have dimension at most $n-7$.

[TABLE]

The key to proving both Theorems is the observation that Naber-Valtorta’s technique gives the following linear interior bound on the singular set: if $T$ is area-minimizing in $U\subset\mathbb{R}^{n+1}$, with $\partial T\llcorner U=0$, then for every $\epsilon>0$, we have:

[TABLE]

For the Neumann problem (Theorem 0.2), we can adapt the techniques of [NV15] to prove a priori estimates on the singular set in a neighborhood of the barrier. Unfortunately, it’s not clear that a good Dirichlet boundary version of Naber-Valtorta exists, in any more generality than is considered in Theorem 0.1. The problem is that there is not necessarily a good relationship between regularity and symmetry. If there exists a singular, minimizing hypersurface with Euclidean area growth and linear boundary, then by [HS79] any blow-down sequence would preclude an inclusion like $\mathrm{sing}T\subset S^{n-7}_{\epsilon}$ (here $S^{n-7}_{\epsilon}$ being the $(n-7,\epsilon)$-strata of [CN13]).

Instead, for Theorem 0.1, we can prove an effective version of [HS79], which says that the singular set is some uniform distance away from the boundary curve. It’s tempting to think an ineffective, quantitative version of [HS79] might hold for more general Dirichlet setups, but the problem is the same as above.

[TABLE]

for any $S\in\mathcal{I}_{n}(\mathbb{R}^{n+1})$, $\partial S=0$, $\mathrm{spt}S\subset B_{r}(x)$, and some fixed $c_{0}$. Suppose $\partial T=[S]$ is an oriented, embedded, multiplicity-one $C^{1,\alpha}$-manifold $S$, and suppose there is a $C^{1,\alpha}$ domain $\Omega$ so that $\mathrm{spt}T\subset\overline{\Omega}$, $S\subset\partial\Omega$. Then

[TABLE]

The same proof works, using [DS02], [Bom82] in place of [HS79], [All72], and a minor modification of [NV15].

The following examples illustrates some of the problems in extending our proof of Theorem 0.1 to more general settings.

The half-helicoid structure could be seen locally for finite $S$, if one does not assume a priori area bounds on $S$. For example, one can imagine a connected boundary curve $S$, which is composed of line segment $L$, and a curve that wraps around $L$ many times. By taking the wrapping curve to go further and further out, one can arrange $S$ to satisfy $r_{1,\alpha}(S)\geq 1$, but take the separation along $L$ of the wrappings to zero. The minimizing integral current $T$ spanning $S$ will look very much like a compressed half-helicoid near the line segment. We cannot decompose this $T$ near $L$ into pieces of uniformly bounded area.

I thank Otis Chodosh for several illuminating conversations, and pointing out the half-helicoid is area-minimizing. This work was supported in part by NSF grant DMS-1606492.

0.5. Proof of Theorem 0.1

The following quantifies Hardt-Simon’s boundary regularity.

Lemma 0.6**.**

There is a constant $\epsilon_{1}(n,\alpha)$, so that the following holds. Let $T$ and $S$ be as in Theorem 0.1. Then for all $x\in\mathrm{sing}T$, we have

[TABLE]

Proof.

Towards a contradiction, suppose we have minimizing currents $T_{i}$, with boundary curves $S_{i}$, each contained the boundary of the convex set $\Omega_{i}$, and $x_{i}\in\mathrm{sing}T_{i}$, and $y_{i}\in S_{i}$, so that

[TABLE]

By the maximum principle, $\mathrm{spt}T_{i}\subset\overline{\Omega_{i}}$, and by [HS79], $\mathrm{dist}(x_{i},S_{i})>0$. Since $r_{1,\alpha}(S,y)\leq r_{1,\alpha}(S,y_{i})/2$ for $y\in B_{r_{1,\alpha}(S,y)/2}(y)$, there is no loss in generality in assuming that $y_{i}$ realizes the distance in $S$ to $x_{i}$.

After a rotation, dilation, translation, we can assume $y_{i}=0$, $r_{1,\alpha}(S_{i},y_{i})=2$, and $e_{1}$ is a choice of vector so that $\Omega_{i}\subset\{x:x\cdot e_{1}<0\}$. Moreover, we can take $S_{i}\cap B_{1}$ to be the graph of a function $u_{i}$, define on the line $L=\{x_{1}=x_{n+1}=0\}$, with $|u_{i}|_{1,\alpha}\leq 1$. Notice that $\mathrm{dom}(u_{i})\supset B_{1/2}\cap L$, and that $x_{i}\to 0$. Let us assume $x_{i}\in B_{1/2}$ for all $i$.

Let $h(t,x):[-1,1]\times(L\cap B_{1/2})\to\mathbb{R}^{n+1}$ be defined as

[TABLE]

and let $R_{i}=(h_{i})_{\sharp}([-1,1]\times[L\cap B_{1/2}])$. Then, as an element of $\mathcal{I}_{n}(B_{1/2})$, $\partial R_{i}=[S_{i}]\llcorner B_{1/2}$. In particular, $T_{i}-R_{i}\in\mathcal{I}_{n}(B_{1/2})$ has no boundary. By standard decomposition of codimension-one currents [Sim83], we can find open sets $E_{i,j}\subset E_{i,j+1}\subset\ldots B_{1/2}$, so that $[E_{i,j}]\in\mathcal{I}_{n}(B_{1/2})$ satisfies:

[TABLE]

Since $(T_{i}-R_{i})\llcorner\Omega^{c}=(-R_{i})\llcorner\Omega^{c}$, and the $E_{i,j}$ are nested, we have $\mathrm{spt}\partial[E_{i,j}]\subset\overline{\Omega}$ for all but one $j=j_{i}$. Therefore, we have

[TABLE]

and hence $\partial[E_{i,j_{i}}]+R_{i}$ and each $\partial[E_{i,j}]$ ($j\neq j_{i}$) are area-minimizing. By volume comparison agains balls, and the estimate $||R_{i}||(B_{r})\leq c(n)r^{n}$, we get that

[TABLE]

We break into two cases. First, assume that $x_{i}\in\mathrm{spt}(\partial[E_{i,j_{1}}]+R_{i})$ for all $i$. Let $\lambda_{i}=|x_{i}|^{-1}$, and consider the dilates

[TABLE]

So that $T^{\prime}_{i}$ has a singularity at distance $1$ from $\lambda_{i}S_{i}$.

We can pass to a subsequence (also denoted $i$), so that $(\eta_{\lambda_{i}})_{\sharp}[E_{i,j_{1}}]\to[E]$, for some open set $E$. Since $\lambda_{i}S_{i}\to L$ in $C^{1,\alpha}$, we have $(\eta_{\lambda_{i}})_{\sharp}R_{i}\to[H]$, where $H=\{x_{n+1}=0,x_{1}>0\}$ and $[H]$ is endowed with the orientation so that $\partial[H]=L$.

In particular, we have $T^{\prime}_{i}\to T=\partial[E]+[H]$, where $\partial T=[L]$. Since each $T^{\prime}_{i}$ is minimizing, $T$ is minimizing also, and $T_{i}^{\prime}$ converge as both currents and measures. By construction, $T$ has a singularity at distance $1$ from $L$, $T$ has Euclidean volume growth, and $\mathrm{spt}T\subset\{x:x\cdot e_{1}\leq 0\}$.

Since $T$ is minimizing with Euclidean volume growth, we can take a tangent cone $C$ at infinity (as both currents and variflds). $C$ satisfies $\partial[C]=[L]$, and so by [HS79] $C$ is planar. Since we can write $C=\partial[F]+[H]$ for some open set $F$, and $\mathrm{spt}C\subset\{x:x\cdot e_{1}\leq 0\}$, in fact $C$ must be a multiplicity-one half-plane. By monotonicity we must have that $T$ is a multiplicity-one half-plane also, and hence $T$ is regular. This is a contradiction.

We are left with the second case: for all $i$, $x_{i}\in\mathrm{spt}\partial[E_{i,j}]$ for some $j\neq j_{i}$. Write $E_{i}=E_{i,j}$ for the open set, for which $x_{i}\in\mathrm{spt}\partial E_{i}$. Consider the dilates $E^{\prime}_{i}=\lambda_{i}E_{i}$. Then we can pass to a subsequence, to get convergence as currents $[E^{\prime}_{i}]\to[E]$, convergence as currents and measures $\partial[E^{\prime}_{i}]\to\partial[E]$, for $0\neq[E]\in\mathcal{I}_{n+1}(\mathbb{R}^{n+1})$ satisfying: a) $\partial[E]$ is minimizing; b) $\partial[E]$ has a singularity at distance $1$ from the origin; and c) $E\subset\{x:x\cdot e_{1}\leq 0\}$.

Properties a), c) imply that any tangent cone at infinity of $\partial[E]$ is a multiplicity-one plane, and hence $\partial[E]$ is a multiplicity-one plane. This contradicts property b), and therefore completes the proof of Lemma 0.6. ∎

Lemma 0.7**.**

Let $T\in\mathcal{I}_{n}(B_{1})$ be area-minimizing, with $\partial T=0$. Then we have

[TABLE]

Proof.

We can decompose $T=\sum_{i}\partial[E_{i}]$, for $E_{i}\subset E_{i+1}\subset\ldots\subset B_{1}$, so that $||T||=\sum_{i}||\partial[E_{i}]||$, and hence each $\partial[E_{i}]\in\mathcal{I}_{n}(B_{1})$ is minimizing in $B_{1}$.

Since $||\partial[E_{i}]||(B_{1})\leq||\partial[E_{i}\cup B_{3/4}]||(B_{1})$, we have $||\partial[E_{i}]||(B_{3/4})\leq c(n)$. On the other hand, by monotonicity, if $\mathrm{spt}\partial[E_{i}]\cap B_{1/2}\neq\emptyset$, then $||\partial[E_{i}]||(B_{1})\geq 1/c(n)$. From the estimates of [NV15], we have

[TABLE]

We can sum up contributions:

[TABLE]

Proof of Theorem 0.1.

By scaling, there is no loss in assuming $r_{1,\alpha}(S)=1$. Lemma 0.6 implies that $B_{\epsilon}(S)\cap\mathrm{sing}T=\emptyset$, where $\epsilon=\epsilon_{1}(n,\alpha)$.

Let $\{x_{j}\}_{j}$ be a maximal $(\epsilon/2)$-net in $\mathrm{spt}T\setminus B_{\epsilon}(S)$. Then the balls $\{B_{\epsilon/2}(x_{j})\}_{j}$ cover $\mathrm{spt}T\setminus B_{\epsilon}(S)$, and the balls $\{B_{\epsilon}(x_{j})\}_{j}$ have overlap bounded by $c(n)$. For each $j$, $\partial T\llcorner B_{\epsilon}(x_{j})=0$, and so by Lemma 0.7 we have

[TABLE]

Using bounded overlap of the $\{B_{\epsilon}(x_{j})\}_{j}$, and the isoperimetric inequality due to [FJA86], we deduce that

[TABLE]

0.8. Proof of Theorem 0.2

We will show that the arguments of [NV15], [Gru87], and [GJ86] prove the following: there is an $\epsilon=\epsilon(n)$, so that for $x\in\mathrm{spt}T\cap\partial\Omega$, and $r=r_{1,1}(\partial\Omega)$, we have

[TABLE]

Given this estimate, the bound of Theorem 0.2 follows by a straightforward covering argument as in the proof of Theorem 0.1.

By scaling, we can and shall assume that $r_{1,1}(\partial\Omega)=1/\Gamma$, for $\Gamma\leq\epsilon_{2}(n)$ chosen sufficiently small so that in $B_{4}(\partial\Omega)$ the nearest-point projection $\xi(x)$ to $\partial\Omega$ is well-defined and satisfies $|\xi|_{C^{1}}\leq 1$. Define the reflection function $\sigma(x)=2\xi(x)-x$, and the linear reflection $i_{x}$ about $T_{\xi(x)}\partial\Omega$.

Take $T\in\mathcal{I}_{n}(B_{2})$ area-minimizing with free-boundary in $\Omega$. Define $T^{\prime}=T-\sigma_{\sharp}T$, so that $\partial T^{\prime}=0$. Then we can decompose $T^{\prime}$ as

[TABLE]

for nested open sets $E_{i}\subset E_{i+1}$. Moreover, since $T^{\prime}\llcorner\Omega=T$ we can write

[TABLE]

From (4), we get that each $\partial[E_{i}]\llcorner\Omega$ is area-minimizing, with free-boundary in $\Omega$. By comparison against $\partial[E_{i}\cup B_{r}(x)]\llcorner\Omega$, we have the a priori mass bounds

[TABLE]

Additionally, [Gru87] showed $T^{\prime}$ admits a certain almost-minimizing property, in the following sense:

[TABLE]

for every $S\in\mathcal{I}_{n}(B_{2})$ with $\partial S=0$, $\mathrm{spt}S\subset B_{r}(x)$, and every $B_{r}(x)\subset B_{2}$ with $x\in\partial\Omega$.

[GJ86] define the following monotonicity. For $x\in B_{1}$, and $r<1-|x|$, let

[TABLE]

Notice that when $\Omega$ is a half-space, then $\tilde{\theta}_{T}(x,r)=\theta_{T^{\prime}}(x,r)$, and in general we have $\tilde{\theta}_{T}(x,r)=\theta_{T}(x,r)$ when $r<\mathrm{dist}(x,\partial\Omega)$. Here $\theta_{T}(x,r)=r^{-n}||T||(B_{r}(x))$ for the usual Euclidean density ratio, and $\theta_{T}(x)=\lim_{r\to 0}\theta_{T}(x,r)$ whenever it exists.

For $0<s<r<1-|x|$, [GJ86] prove

[TABLE]

Monotonicity (6) implies that the density $\tilde{\theta}_{T}(x)=\lim_{r\to 0}\tilde{\theta}_{T}(x,r)$ is a well-defined, upper-semi-continuous function on $B_{1}$, which is $\geq 1$ on $\mathrm{spt}T$.

The above discussion, and the works of [Gru87], [GJ86], give:

Lemma 0.9**.**

Let $\Omega_{i}$ be a sequence of $C^{2}$ domains, with $r_{1,1}(\partial\Omega_{i}\cap B_{2})\to\infty$, and $T_{i}\in\mathcal{I}_{n}(B_{2})$ a sequence of area-minimizing currents with free-boundary in $\Omega_{i}$. Suppose $T_{i}\to T$. Then

1. (1)

$T$* is area-minimizing, with free-boundary in a half-space, and $||T_{i}||\to||T||$.* 2. (2)

$T^{\prime}_{i}\to T^{\prime}$* as currents and measures, and $\tilde{\theta}_{T_{i}}(x,r)\to\theta_{T^{\prime}}(x,r)$ for all $x\in B_{2}$, and a.e. $0<r<2-|x|$. Here $T_{i}^{\prime}=T_{i}-(\sigma_{i})_{\sharp}T_{i}$, where $\sigma_{i}$ is the reflection function associated to $\Omega_{i}$.* 3. (3)

If $x_{i}\to x\in B_{2}$, and $r_{i}\to 0$, then $\limsup_{i}\tilde{\theta}_{T_{i}}(x_{i},r_{i})\leq\tilde{\theta}_{T}(x)=\theta_{T^{\prime}}(x)$. 4. (4)

If $T^{\prime}$ is regular, then the $T_{i}^{\prime}\llcorner B_{1}$ are regular for $i$ sufficiently large.

Proof.

Since $\sigma_{i}\to\sigma$ in $C^{1}$, we have $T_{i}^{\prime}\to T^{\prime}$. The convergence of measures $||T_{i}^{\prime}||\to||T^{\prime}||$ is a standard argument using the almost-minimizing property (6). Convergence $||T_{i}||\to||T||$ then follows from the fact that $T_{i}^{\prime}\llcorner\overline{\Omega_{i}}=T_{i}$.

Convergence of the $\tilde{\theta}_{T}$ follows because we can estimate

[TABLE]

where $\kappa_{i}\to 0$ as $i\to\infty$, and because $||T||(\partial B_{r}(x))=0$ for a.e. $r$. Upper-semi-continuity follows by convergence of $\tilde{\theta}_{T}(x,r)$, and monotonicity.

The last property (4) is a direct consquence of the decomposition (4) and the Allard-type regularity theory of [GJ86]. ∎

We show the following variant of [NV15] (recall that $r_{1,1}(\partial\Omega)=1/\Gamma$).

Theorem 0.10** (compare from [NV15]).**

There is an $\epsilon_{3}=\epsilon_{3}(n,\Lambda)$, so that if $T\in\mathcal{I}_{n}(B_{2})$ is area-minimizing, with free-boundary in $\Omega$, and $||T||\leq\Lambda$, and $\Gamma\leq\epsilon_{3}$, then

[TABLE]

When $T=\partial[E]$, we get $\Lambda=c(n)$, and then using the decomposition (4) in an identical argument to Lemma 0.7, we deduce the required (3).

The argument of [NV15] requires only the monotonicity formula (6), and the following two theorems, which are essentially Lemmas 7.2, 7.3 and Theorem 6.1 in [NV15] (or Lemma 3.1, Theorem 5.1 in [EE17]). The rest of [NV15] is entirely general (see e.g. [Ede]).

Theorem 0.11**.**

There is an $\eta_{0}=\eta_{0}(n,\alpha,\Lambda,\gamma,\rho)$, so that the following holds. Take $B_{6r}(x)\subset B_{2}$. Let $T\in\mathcal{I}_{n}(B_{6r}(x))$ be an area-minimizer with free-boundary in $\Omega$, and take $\eta\leq\eta_{0}$. Suppose

[TABLE]

then at least one of the following occurs:

1. (1)

we have

[TABLE] 2. (2)

there is an affine $(n-8)$-space $p+L^{n-8}$, so that

[TABLE]

Theorem 0.12**.**

There is a $\delta(n,\alpha,\Lambda)$ so that the following holds. Take $B_{10r}(x)\subset B_{2}$. Let $T\in\mathcal{I}_{n}(B_{10r}(x))$ be an area-minmizer with free-boundary in $\Omega$, and $\mu$ a finite Borel measure. Suppose that

[TABLE]

Then we have

[TABLE]

where the infimum is over affine $(n-7)$-planes $p+L^{n-7}$.

Proof of Theorem 0.11.

The proof consists of two contradiction arguments, verbatim to Theorem 5.1 in [EE17]. In place of the $\epsilon$-strata, we use the following consequence of Lemma 0.9: Suppose $T_{i}\in\mathcal{I}_{n}(B_{6})$ is a sequence of area-minimizers with free-boundary in $\Omega_{i}$, so that $r_{1,1}(\partial\Omega_{i}\cap B_{6})\to\infty$ and $T_{i}\to T$. If $T^{\prime}\llcorner B_{2}$ coincides with a cone, that is invariant along an $(n-6)$-space, then $T_{i}\llcorner B_{1}$ is regular for sufficiently large $i$. ∎

Proof of Theorem 0.12.

The proof divides into two parts, which are verbatim to Lemma 6.2 and Proposition 6.6 in [NV15] (or Theorem 5.1 in [EE17]). The first part is a direct consequence of the monotonicity formula (6). The second part is a straightforward contradiction argument. The proof in [NV15] uses varifold convergence. For integral currents, one can use the fact for any $(n+1)$-form $\omega$ and any vector $v$, we have

[TABLE]