Explicit Determination in ${\Bbb R}^{N}$ of $(N-1)$-Dimensional Area Minimizing Surfaces with Arbitrary Boundaries

TL;DR

This paper introduces a finite-time algorithm to explicitly approximate area-minimizing surfaces in ${f R}^N$ with arbitrary boundaries, addressing a long-standing computational challenge in geometric measure theory.

Contribution

The paper presents the first explicit, finite-time algorithm to compute near-minimal area surfaces spanning arbitrary boundaries in ${f R}^N$, with rigorous approximation guarantees.

Findings

Algorithm computes an integral current close to the minimal surface.

Guarantees on Hausdorff and flat norm distances.

Approximation within any specified tolerance $ ext{epsilon}$.

Abstract

Let $N\ge3$ be an integer and $B$ be a smooth, compact, oriented, $(N-2)$-dimensional boundary in ${\Bbb R}^{N}$. In 1960, H. Federer and W. Fleming proved that there is an $(N-1)$-dimensional integral current spanning surface of least area. The proof was by compactness methods and non-constructive. In 1970 H. Federer proved the definitive regularity result for such a codimension one minimizing surface. Thus it is a question of long standing whether there is a numerical algorithm that will closely approximate the area minimizing surface. The principal result of this paper is an algorithm that solves this problem. Specifically, given a neighborhood $U$ around $B$ in ${\Bbb R}^{N}$ and a tolerance $\epsilon>0$, we prove that one can explicitly compute in finite time an $(N-1)$-dimensional integral current $T$ with the following approximation requirements: (1) spt$(\partial T)\subset U$. (2) $B$ and $\partial T$ are within distance $\epsilon$ in the Hausdorff distance. (3) $B$ and $\partial T$ are within distance $\epsilon$ in the flat norm distance. (4) ${\mathbb M}(T)<\epsilon+\inf\{{\mathbb M}(S):\partial S=B\}$. (5) Every area minimizing current $R$ with $\partial R=\partial T$ is within flat norm distance $\epsilon$ of $T$.

Equations

$$I(r)=\{x:\mathrm{dist}(x,\mathrm{spt}B)<r\},O(r)=\{x:\mathrm{dist}(x,\mathrm{spt}B)\ge r\}.$$

$$\mathbb{M}(T_i)\le {ϵ}_i+\inf \{\mathbb{M}(T):T\in \mathcal{T}(i)\}.$$

$$\mu -{ϵ}_i\le \mathbb{M}(T_i)\le \mu +2{ϵ}_i$$

$$\mu =\mathbb{M}(R)\le \mathbb{M}(S_i+T_i)\le \mathbb{M}(S_i)+\mathbb{M}(T_i)\le {ϵ}_i+\mathbb{M}(T_i),$$

$$\mathbb{M}(T_i)\le {ϵ}_i+\mathbb{M}(\hat{R})\le 2{ϵ}_i+\mathbb{M}(R)=2{ϵ}_i+\mu ,$$

$$\mathbb{M}[T_i\text{vrule}width=0.59998pt,height=7.5347pt,depth=0.86108pt\text{vrule}height=-0.34444pt,width=3.75pt,depth=0.86108ptI({ϵ}_1)]\le ϵ/2.$$

$$J=\{i:\mathbb{M}[T_i\text{vrule}width=0.59998pt,height=7.5347pt,depth=0.86108pt\text{vrule}height=-0.34444pt,width=3.75pt,depth=0.86108ptI({ϵ}_1)]\le ϵ/2\}.$$

$$\mathbb{M}[T_i\text{vrule}width=0.59998pt,height=7.5347pt,depth=0.86108pt\text{vrule}height=-0.34444pt,width=3.75pt,depth=0.86108ptI({ϵ}_1)]>ϵ/2$$

$$\mathbb{M}[T_i]=\mathbb{M}[T_i\text{vrule}width=0.59998pt,height=7.5347pt,depth=0.86108pt\text{vrule}height=-0.34444pt,width=3.75pt,depth=0.86108ptI({ϵ}_1)]+\mathbb{M}[T_i\text{vrule}width=0.59998pt,height=7.5347pt,depth=0.86108pt\text{vrule}height=-0.34444pt,width=3.75pt,depth=0.86108ptO({ϵ}_1)]$$

$$\mathbb{M}[T_i\text{vrule}width=0.59998pt,height=7.5347pt,depth=0.86108pt\text{vrule}height=-0.34444pt,width=3.75pt,depth=0.86108ptO({ϵ}_1)]=\mathbb{M}[T_i]-\mathbb{M}[T_i\text{vrule}width=0.59998pt,height=7.5347pt,depth=0.86108pt\text{vrule}height=-0.34444pt,width=3.75pt,depth=0.86108ptI({ϵ}_1)]<\mathbb{M}[T_i]-ϵ/2.$$

$$\underset{i\to \infty }{\lim }\mathbb{M}[T_i\text{vrule}width=0.59998pt,height=7.5347pt,depth=0.86108pt\text{vrule}height=-0.34444pt,width=3.75pt,depth=0.86108ptO({ϵ}_1)]\le \mu -ϵ/2,$$

$$\Vert R\Vert [O({ϵ}_1)]\le \mu -ϵ/2$$

$$\Vert R\Vert [I({ϵ}_1)]\ge ϵ/2,$$

$$T_i^{\prime }=T_i\text{vrule}width=0.59998pt,height=7.5347pt,depth=0.86108pt\text{vrule}height=-0.34444pt,width=3.75pt,depth=0.86108pt\mathcal{K}\mathrm{and}{B}_i=\partial T_i^{\prime }.$$

$$\mathbb{M}(Q_i)\le {ϵ}_i+\inf \{\mathbb{M}(Q):Q\in \mathcal{Q}(i)\}.$$

$$B=\partial S_1+\partial T_{i_0},{\mathrm{dist}}_H[\mathrm{spt}(S_1),\mathrm{spt}(B)]<{ϵ}_{i_0},\mathrm{and}\mathbb{M}(S_1)<{ϵ}_{i_0}.$$

$$B$$

$$\mathrm{spt}(S_1+T_{i_0}\text{vrule}width=0.59998pt,height=7.5347pt,depth=0.86108pt\text{vrule}height=-0.34444pt,width=3.75pt,depth=0.86108pt{\mathbb{R}}^N\setminus \mathcal{K})\subset U$$

$$\mathbb{M}\Big{(}S_{1}+T_{i_{0}}\kern 1.99997pt\vrule width=0.59998pt,height=7.5347pt,depth=0.86108pt\vrule height=-0.34444pt,width=3.75pt,depth=0.86108pt\kern 1.99997pt\mathbb{R}^{N}\setminus\mathcal{K}\Big{)}\leq\mathbb{M}(S_{1})+\mathbb{M}(T_{i_{0}}\kern 1.99997pt\vrule width=0.59998pt,height=7.5347pt,depth=0.86108pt\vrule height=-0.34444pt,width=3.75pt,depth=0.86108pt\kern 1.99997pt\mathbb{R}^{N}\setminus\mathcal{K})\leq\epsilon_{i_{0}}+\epsilon/2\leq\epsilon\,,$$

$$\mathbb{M}(T_{i_0}^{\prime })<ϵ+\inf \{\mathbb{M}(S):\partial S=B\}.$$

$$\mathbb{M}(R)\le \mathbb{M}(T_{i_0}^{\prime }).$$

$$\mathbb{M}(\hat{R})\le \mathbb{M}(R)+{ϵ}_{i_0}\le \mathbb{M}(T_{i_0}^{\prime })+{ϵ}_{i_0}$$

$$\mathbb{M}(Q_{i_0})\le {ϵ}_{i_0}+\mathbb{M}(\hat{R})\le \mathbb{M}(T_{i_0}^{\prime })+2{ϵ}_{i_0},$$

$$\mathbb{M}(W)<ϵ/2.$$

$$\mathbb{M}(X)+\mathbb{M}(Y)\le {ϵ}_{i_0}.$$

$$T_{i_0}^{\prime }-R=S_2+X+\partial Y+\partial W,$$

$$\mathbb{M}(S_2)+\mathbb{M}(X)+\mathbb{M}(Y)+\mathbb{M}(W)\le 2{ϵ}_{i_0}+ϵ/2\le ϵ.$$

$$\mathbb{M}(Q_i)<\mathbb{M}(T_i^{\prime })+3{ϵ}_i$$

$$P_i=S_i+T_i\text{vrule}width=0.59998pt,height=7.5347pt,depth=0.86108pt\text{vrule}height=-0.34444pt,width=3.75pt,depth=0.86108pt{\mathbb{R}}^N\setminus \mathcal{K}+S_i^{\prime }+Q_i.$$

$$\partial P_i$$

Full text

Specifically, given a neighborhood $U$ around $B$ in $\mathbb{R}^{N}$ and a tolerance $\epsilon>0$, we prove that one can explicitly compute in finite time an $(N-1)$-dimensional integral current $T$ with the following approximation requirements:

1. (1)

$\operatorname{spt}(\partial T)\subset U$. 2. (2)

$B$ and $\partial T$ are within distance $\epsilon$ in the Hausdorff distance. 3. (3)

$B$ and $\partial T$ are within distance $\epsilon$ in the flat norm distance. 4. (4)

$\mathbb{M}(T)<\epsilon+\inf\{\mathbb{M}(S):\partial S=B\}$. 5. (5)

Every area minimizing current $R$ with $\partial R=\partial T$ is within flat norm distance $\epsilon$ of $T$.

The work of the second author was supported in part by a grant from the National Science Foundation.

1. Introduction

In this paper, we will follow the notation and terminology of Federer [Fed69] except as otherwise noted. Fix a positive integer $N\geq 3$. In 1960, H. Federer and W. Fleming [FF60] proved that for any smooth, compact, $(N-2)$-dimensional, oriented boundary in $\mathbb{R}^{N}$, there is an $(N-1)$-dimensional spanning surface of least area. The proof was by compactness methods and non-constructive. In 1970 H. Federer [Fed70] proved the definitive regularity result for such a codimension one minimizing surface. Thus it is a question of long standing whether there is a numerical algorithm that will closely approximate the area minimizing surface. The principal result of this paper is an algorithm that solves this problem:

1.1 Theorem** (Main Result).**

Given a smooth $(N-2)$-dimensional integral boundary $B$, neighborhood $U$ around $B$, and $\epsilon>0$, we will compute in finite time an integral current $T$ that we can guarantee satisfies the following requirements:

1. (1)

$\operatorname{spt}(\partial T)\subset U$. 2. (2)

$\operatorname{dist}_{H}[\operatorname{spt}(\partial T),\operatorname{spt}(B)]<\epsilon$, where $\operatorname{dist}_{H}[\cdot,\cdot]$ is Hausdorff distance. 3. (3)

$\mathcal{F}(\partial T-B)<\epsilon$. 4. (4)

$\mathbb{M}(T)<\epsilon+\inf\{\mathbb{M}(S):\partial S=B\}$. 5. (5)

Every area minimizing current $R$ with $\partial R=\partial T$ is within flat norm distance $\epsilon$ of $T$.

1.2 Remarks*.*

We should note that there is a limit to what can be expected. For general boundary curves, the best reasonable result is the approximation, in both area and location, of an area minimizing surface that has boundary near the given boundary and has area nearly equal to the minimum of areas of surfaces spanning the given boundary.

• •

In general, there will be little a priori control of the topology of a minimizing surface.

• •

In general, the area minimizing surface with a given boundary is not unique. Even though F. Morgan [Mor81] has shown that for a generic boundary the area minimizing surface is unique, there are but few situations in which uniqueness can be guaranteed a priori.

• •

Distinct small perturbations of the boundary can result in unique area minimizing surfaces that are widely separated even though their boundaries are nearly identical. It was noted by M. Beeson [Bee77] that a consequence of such discontinuous behavior is that, in a certain formal system, the area minimizing surface is not computable. Thus we believe that it is essential to seek an approximation to an area minimizing surface the boundary of which is near to, but not necessarily identical to, the given boundary.

The last two items above concerning uniqueness and non-uniqueness present the crucial difficulties in closely approximating the location of an area minimizing surface, because a surface of nearly minimum area for the given boundary may be far away in location from any area minimizer for that boundary. We deal with these difficulties by using a sequence of more and more precise approximations in which we first construct a surface $T_{i}$ of nearly minimum area, and then second consider an auxiliary minimization problem. This auxiliary problem seeks the minimum area among surfaces satisfying two constraints which we describe informally as follows. The first constraint is that the boundary of each of the surfaces considered must equal the boundary of an appropriate portion, $T^{\prime}_{i}$, of the surface $T_{i}$. The second constraint is that each of the surfaces must be relatively far from $T^{\prime}_{i}$ in the flat norm.

Continuing our informal discussion, if $\epsilon>0$ is specified at the outset and if the parameters defining large and small and near and far are chosen correctly vis-à-vis that $\epsilon$, then in the above sequence of constructions and minimizations, it eventually must happen both that $T^{\prime}_{i}$ differs little from $T_{i}$ and that the minimum area among the surfaces considered in the auxiliary problem is relatively large. Consequently, the $T^{\prime}_{i}$ constructed at that iteration is such that all surfaces relatively far from $T^{\prime}_{i}$ have relatively large area. Thus the surfaces with relatively small area all must be relatively near to $T^{\prime}_{i}$.

In the previous papers [Par77] and [Par86], the theoretical basis was developed for computing approximations to area minimizing surfaces by numerically approximating functions of least gradient. Those papers required that the given boundary for which an area minimizing spanning surface was sought must lie on the surface of a convex set. An important feature of the results in those papers was that one could be certain, at least in principle, of when sufficient computation had been done to guarantee any desired accuracy of the approximation in the sense of Hausdorff distance.

The method described in [Par77] and [Par86] was implemented numerically in [Par92]. The results reported there and later results in [Par93] showed that, in practice, the method gives much better approximations than the theorems of [Par77] and [Par86] guarantee.

The requirement of [Par77] and [Par86] that the boundary lie on the surface of a convex set is often not met. Various alternative methods are available for application in these circumstances. These are developed in the extremely general covering space approach of K. Brakke [Bra95b], in the duality approach in the thesis of J. Sullivan [Sul90], the more general work of K. Brakke [Bra95a], and in the modification of the least gradient method in our previous work [PP96] and [PP97]. The results of [Sul90] and [Bra95a] provide a way to approximate the area of the area minimizing surface (but not the position), and implicitly so do the results of [PP97].

We dedicate this paper to the memory of our thesis advisor and friend Frederick J. Almgren, Jr.

2. The Algorithm

The Approximation Theorem obtained by Federer and Fleming tells us that any integral current can be approximated arbitrarily well by an integral polyhedral chain. Consequently, given a smooth, compact, embedded, $(N-2)$-dimensional boundary in $\mathbb{R}^{N}$, an area minimizing surface spanning the given boundary can be obtained as the limit of integral polyhedral chains obtained by minimizing mass in an increasing family of finite dimensional subspaces of the vector space of $(N-1)$-dimensional polyhedral chains, $\mathcal{P}_{N-1}(\mathbb{R}^{N})$. As a computational method, the obvious shortcoming of such an approach is that, if one has in mind a desired level of accuracy of approximation, there is no way to know whether one has achieved it. What is lacking is a priori information on which finite dimension subspace of $\mathcal{P}_{N-1}(\mathbb{R}^{N})$ is required to obtain the desired accuracy of approximation.

In his thesis [Sul90], John Sullivan has addressed this lack of a priori information. Sullivan’s approximation is carried out using an appropriate cell complex obtained by slicing space with equally spaced parallel planes in each of many directions, a structure that he calls a “multigrid.”

2.1 Definition**.**

A multigrid in $\mathbb{R}^{N}$ is the set of chains generated by a finite family of convex polyhedra in $\mathbb{R}^{N}$ and by their vertices, edges, and faces. In our implementation, we need include only the $(N-1)$-dimensional faces and $(N-2)$-faces.

Sullivan’s approximation result is the following:

2.2 Theorem** (Sul90, Theorem 6.1).**

Given $\epsilon$ and an $(N-1)$-current $T$, we can pick a multigrid $C$ such that $T$ has a good approximation $S$, which is a chain in $C$, is flat close to $T$, and has not much more mass, $\mathbb{M}(S)\leq(1+\epsilon)\,\mathbb{M}(T)$. In fact the choice of $C$ can be made merely knowing $\epsilon$ and bounds on $\mathbb{M}(T)$ and on the mass of its boundary.

Using this last approximation result, Sullivan obtains the next result (which we paraphrase) regarding an algorithm for approximating the minimum area that is required to span a given boundary cycle.

2.3 Theorem** (Sul90, Corollary 6.2).**

Given any boundary cycle in $\mathbb{R}^{N}$, with some a priori lower bound on the area of a possible area-minimizing surface, a surface with no more than $1+\epsilon$ times the true minimum area can be found by solving a linear programming problem.**

In the statement of Theorem 2.3, Sullivan focuses on the approximation of the minimum area. But we note that in Theorem 2.2 the approximating surface also approximates the given boundary; a fact that is important in our work. By making use of the top-dimensional polyhedra in a sequence of finer and finer multltigrids, we are able to obtain an algorithm that not only approximates the minimum area, but that also approximates both the area and the location (in the sense of the $\mathcal{F}$-norm) of an area minimizer with boundary nearly equal to the given boundary. This algorithm is the first to accomplish that goal.

2.4 Theorem**.**

Let $B\in\mathbb{I}_{N-2}$ with $\partial B=0$ and smooth support be given. Let $\epsilon>0$ be given. Let an open set, $U$, with $\operatorname{spt}(B)\subset U$ be given. Then there is a computation requiring finitely many multigrid minimizations that results in a $T$ guaranteed to satisfy the following requirements:

1. (1)

$\operatorname{spt}(\partial T)\subset U$, 2. (2)

$\operatorname{dist}_{H}[\,\operatorname{spt}(\partial T),\,\operatorname{spt}(B)\,]<\epsilon$, 3. (3)

$B=\partial S+\partial T$* with $\operatorname{spt}(S)\subset U$ and $\mathbb{M}(S)<\epsilon$,* 4. (4)

$\mathbb{M}(T)<\epsilon+\inf\{\mathbb{M}(S):\partial S=B\}$, 5. (5)

every area minimizing current $R$ with $\partial R=\partial T$ is within $\mathcal{F}$-distance $\epsilon$ of $T$.

Proof. Let $B\in\mathbb{I}_{N-2}$ with $\partial B=0$ and smooth support be given. Let $\epsilon>0$ be given. Let the open set $U$ with $\operatorname{spt}(B)\subset U$ be given.

For each $0<r$, set

[TABLE]

Let $0<\epsilon_{i}$, $i=1,2,\dots$, be a decreasing sequence with limit [math]. Choose $\epsilon_{1}$ so that

• •

$\epsilon_{1}<\epsilon/4$,

• •

$\hbox{\rm Clos}[I(2\,\epsilon_{1})]\subset U$,

• •

$\|R\|[I(\epsilon_{1})]<\epsilon/3$ holds for any mass minimizer with $\partial R=B$, which we can do by Proposition 5.6 of [Sul90].

For each $i$, use Sullivan’s approximation method (Theorem 2.2) to form a multigrid $\mathcal{G}(i)$ such that for any mass minimizer $R$ with $\partial R=B$ there exists $\widehat{R}\in\mathcal{G}(i)\cap\mathcal{P}_{2}$ such that

• •

there exists $S$ with $B=\partial S+\partial\widehat{R}$, $\operatorname{dist}_{H}[\,\operatorname{spt}(S),\,\operatorname{spt}(B)\,]<\epsilon_{i}$, and $\mathbb{M}(S)<\epsilon_{i}$,

• •

$\operatorname{dist}_{H}[\,\operatorname{spt}(\partial\widehat{R}),\,\operatorname{spt}(B)\,]<\epsilon_{i}$,

• •

$\operatorname{spt}(\partial\widehat{R})\subset U$,

• •

$\mathbb{M}(\widehat{R})\leq\mathbb{M}(R)+\epsilon_{i}$.

Choose the multigrids so that $\mathcal{G}(1)\subset\mathcal{G}(2)\subset\mathcal{G}(3)\subset\cdots$.

For each $i$, let $\mathcal{T}(i)\subset\mathcal{G}(i)\cap\mathcal{P}_{N-1}$ be the set of currents, $T$, satisfying

• •

there exists $S$ with $B=\partial S+\partial T$, $\operatorname{dist}_{H}[\,\operatorname{spt}(S),\,\operatorname{spt}(B)\,]<\epsilon_{i}$, and $\mathbb{M}(S)<\epsilon_{i}$,

• •

$\operatorname{dist}_{H}[\,\operatorname{spt}(\partial T),\,\operatorname{spt}(B)\,]<\epsilon_{i}$,

• •

$\operatorname{spt}(\partial T)\subset U$.

Using an appropriate algorithm, obtain $T_{i}\in\mathcal{T}(i)$ such that

[TABLE]

(We are solving a linear programming problem. We are also not requiring the exact solution; only that we be within $\epsilon_{i}$ of the minimum value of the objective function.)

Claim 1. If $\mu$ denotes the mass of any mass minimizer $R$ with $\partial R=B$, then

[TABLE]

holds for each $i$, and the limit of any $\mathcal{F}$-convergent subsequence of \big{\{}T_{i}\big{\}}_{i=1}^{\infty} is a mass minimizer with boundary equal to $B$.

Proof of Claim. Let $R$ be a mass minimizer with $\partial R=B$.

Since $T_{i}\in\mathcal{T}(i)$, there exists $S_{i}$ with $B=\partial S_{i}+\partial T_{i}=\partial(S_{i}+T_{i})$ and $\mathbb{M}(S_{i})<\epsilon_{i}$. Thus we have

[TABLE]

giving us the left-hand inequality in (1).

We have chosen the multigrid ${\mathcal{G}}(i)$ so that for any mass minimizer $R$ with $\partial R=B$ there exists $\widehat{R}\in\mathcal{T}(i)$ such that

• •

there exists $S$ with $B=\partial S+\partial\widehat{R}$, $\operatorname{dist}_{H}[\,\operatorname{spt}(S),\,\operatorname{spt}(B)\,]<\epsilon_{i}$, and $\mathbb{M}(S)<\epsilon_{i}$,

• •

$\operatorname{dist}_{H}[\,\operatorname{spt}(\partial\widehat{R}),\,\operatorname{spt}(B)\,]<\epsilon_{i}$,

• •

$\operatorname{spt}(\partial\widehat{R})\subset U$,

• •

$\mathbb{M}(\widehat{R})\leq\mathbb{M}(R)+\epsilon_{i}$.

Then $\widehat{R}$ satisfies the conditions for membership in $\mathcal{T}(i)$. By the choice of $T_{i}$, we conclude that

[TABLE]

giving us the right-hand inequality in (1).

Now, let $T^{*}$ be the limit of any $\mathcal{F}$-convergent subsequence of \big{\{}T_{i}\big{\}}_{i=1}^{\infty}. Passing to that subsequence, but without changing notation, we suppose $T_{i}\rightarrow T^{*}$. Letting $S_{i}$ be as above, we have $B=\partial(S_{i}+T_{i})$ and $\mathbb{M}(S_{i})\rightarrow 0$. So $B=\partial T^{*}$. By the lower semicontinuity of mass, $\mathbb{M}(T^{*})\leq\lim_{i\rightarrow\infty}\mathbb{M}(T_{i})=\mu$. Thus $T^{*}$ is a mass minimizer with boundary $B$.

Claim 1 has been proved.

Claim 2. For infinitely many $i$, we have

[TABLE]

Proof of Claim. Suppose Claim 2 were false. Then there would be but finitely many elements in

[TABLE]

Set $i_{0}=1+\max J$. Then

[TABLE]

holds for all $i\geq i_{0}$. Since

[TABLE]

we have

[TABLE]

So

[TABLE]

where, as in Claim 1, $\mu$ denotes the mass of any minimizer with boundary $B$.

Passing to an $\mathcal{F}$-convergent subsequence, but without changing notation, we may suppose $T_{i}$ converges to a mass minimizer $R$ with $\partial R=B$. By the lower semicontinuity of mass,

[TABLE]

holds. Since $\mathbb{M}[R]=\mu$, we have

[TABLE]

contradicting the requirement in the definition of $\epsilon_{1}$ that $\|R\|[I(\epsilon_{1})]<\epsilon/3$ hold.

Claim 2 has been proved.

Let $\mathcal{K}$ be a closed set disjoint from $I(\epsilon_{1}/2)$, containing $O(\epsilon_{1})$, and having a polyhedral boundary. For each $i=1,2,\dots$, set

[TABLE]

For each $i$, use Sullivan’s approximation method (Theorem 2.2) to form a multigrid $\mathcal{G}^{\prime}(i)$, with $\mathcal{G}(i)\subset\mathcal{G}^{\prime}(i)$ and $T^{\prime}_{i}\in\mathcal{G}^{\prime}(i)$, such that for any mass minimizer $R$ with $\partial R=B_{i}$ there exists $\widehat{R}\in\mathcal{G}(i)\cap\mathcal{P}_{N-1}$ such that

• •

there exists $S\in\mathcal{G}^{\prime}(i)\cap\mathcal{P}_{N-1}$ with $B_{i}=\partial S+\partial\widehat{R}$, $\mathbb{M}(S)<\epsilon_{i}$, and $\operatorname{dist}_{H}[\,\operatorname{spt}(S),\,\operatorname{spt}(B_{i})\,]<\epsilon_{i}$,

• •

$\operatorname{dist}_{H}[\,\operatorname{spt}(\partial\widehat{R}),\,\operatorname{spt}(B_{i})\,]<\epsilon_{i}$,

• •

$\operatorname{spt}(\partial\widehat{R})\subset U$,

• •

$\mathbb{M}(\widehat{R})\leq\mathbb{M}(R)+\epsilon_{i}$,

• •

$\widehat{R}-R=X+\partial Y$ for some $X$ and $Y$ with $\mathbb{M}(X)+\mathbb{M}(Y)\leq\epsilon_{i}$.

Choose the multigrids so that $\mathcal{G}^{\prime}(1)\subset\mathcal{G}^{\prime}(2)\subset\mathcal{G}^{\prime}(3)\subset\cdots$.

For each $i$, let $\mathcal{T}^{\prime}(i)\subset\mathcal{G}^{\prime}(i)\cap\mathcal{P}_{N-1}$ be the set of currents, $T$, satisfying

• •

there exists $S\in\mathcal{G}^{\prime}(i)\cap\mathcal{P}_{N-1}$ with $B_{i}=\partial S+\partial T$, $\mathbb{M}(S)<\epsilon_{i}$, and $\operatorname{dist}_{H}[\,\operatorname{spt}(S),\,\operatorname{spt}(B_{i})\,]<\epsilon_{i}$,

• •

$\operatorname{dist}_{H}[\,\operatorname{spt}(\partial T),\,\operatorname{spt}(B_{i})\,]<\epsilon_{i}$,

• •

$\operatorname{spt}(\partial T)\subset U$.

For each $i$, let $\mathcal{Q}(i)\subset\mathcal{T}^{\prime}(i)$ be the set of currents, $Q$, satisfying

• •

$\mathbb{M}(W)\geq\epsilon/2$, where $\partial W=T^{\prime}_{i}-S-Q$ where $S$ is as in the first condition for membership of $Q$ in $\mathcal{T}^{\prime}(i)$.

Notice that if $B_{i}=\partial S+\partial Q$, then $W$ satisfying $\partial W=T^{\prime}_{i}-S-Q$ is unique and $W\in\mathcal{G}^{\prime}(i)\cap\mathcal{P}_{N}$.

Using an appropriate algorithm, obtain $Q_{i}\in\mathcal{Q}(i)$ such that

[TABLE]

(We are solving a linear programming problem. We are also not requiring the exact solution, only that we be within $\epsilon_{i}$ of the minimum value of the objective function.)

Stopping Conditions:

1. (C1)

$\mathbb{M}(Q_{i})\geq\mathbb{M}(T^{\prime}_{i})+3\,\epsilon_{i}$ 2. (C2)

$\mathbb{M}(T\kern 1.99997pt\vrule width=0.59998pt,height=7.5347pt,depth=0.86108pt\vrule height=-0.34444pt,width=3.75pt,depth=0.86108pt\kern 1.99997pt\mathbb{R}^{N}\setminus\mathcal{K})\leq\epsilon/2$

Claim 3. If for some $i_{0}$, the stopping conditions are satisfied, then $T^{\prime}_{i_{0}}$ is the desired approximation. That is,

• •

$B=\partial S+\partial T^{\prime}_{i_{0}}$ with $\operatorname{spt}(S)\subset U$ and $\mathbb{M}(S)<\epsilon$,

• •

$\operatorname{dist}_{H}[\,\operatorname{spt}(\partial T^{\prime}_{i_{0}}),\,\operatorname{spt}(B)\,]<\epsilon$,

• •

$\operatorname{spt}(\partial T^{\prime}_{i_{0}})\subset U$,

• •

$\mathbb{M}(T^{\prime}_{i_{0}})<\epsilon+\inf\{\mathbb{M}(S):\partial S=B\}$,

• •

every mass minimizing current $R$ with $\partial R=\partial T^{\prime}_{i_{0}}=B_{i_{0}}$ is within $\mathcal{F}$-distance $\epsilon$ of $T^{\prime}_{i_{0}}$.

Proof of Claim. By the choice of $\epsilon_{1}$, it is immediate that

• •

$\operatorname{dist}_{H}[\,\operatorname{spt}(\partial T^{\prime}_{i_{0}}),\,\operatorname{spt}(B)\,]<\epsilon$ ,

• •

$\operatorname{spt}(\partial T^{\prime}_{i_{0}})\subset U$

hold.

Since $T_{i_{0}}\in\mathcal{T}(i_{0})$, there exists $S_{1}$ with

[TABLE]

So

[TABLE]

We have

[TABLE]

and

[TABLE]

where we have used the stopping condition (C2).

The right-hand inequality in (1) gives us

[TABLE]

Suppose $R$ is a minimizer with $\partial R=B_{i_{0}}$. Let $\widehat{R}$ be such that

• •

there exists $S_{2}$ with $B_{i_{0}}=\partial S_{2}+\partial\widehat{R}$, $\operatorname{dist}_{H}[\,\operatorname{spt}(S_{2}),\,\operatorname{spt}(B_{i_{0}})\,]<\epsilon_{i_{0}}$, and $\mathbb{M}(S_{2})<\epsilon_{i_{0}}$,

• •

$\operatorname{dist}_{H}[\,\operatorname{spt}(\partial\widehat{R}),\,\operatorname{spt}(B_{i_{0}})\,]<\epsilon_{i_{0}}$,

• •

$\operatorname{spt}(\partial\widehat{R})\subset U$,

• •

$\mathbb{M}(\widehat{R})\leq\mathbb{M}(R)+\epsilon_{i_{0}}$,

• •

$\widehat{R}-R=X+\partial Y$ for some $X$ and $Y$ with $\mathbb{M}(X)+\mathbb{M}(Y)\leq\epsilon_{i_{0}}$.

Notice that the first three conditions above tell us that $\widehat{R}\in\mathcal{T}^{\prime}(i_{0})$.

Next, note that since $R$ is a mass minimizer with $\partial R=\partial T^{\prime}_{i_{0}}$, we have

[TABLE]

Thus

[TABLE]

holds. If it were the case that $\widehat{R}\in\mathcal{Q}(i_{0})$, then the choice of $Q_{i_{0}}$ would give us

[TABLE]

contradicting the stopping condition (C1). We conclude that $\widehat{R}\in\mathcal{T}^{\prime}(i_{0})\setminus\mathcal{Q}(i_{0})$.

Now, let $W$ satisfy $\partial W=T^{\prime}_{i_{0}}-S_{2}-\widehat{R}$ with $S_{2}$ as above. Because $\widehat{R}\notin\mathcal{Q}(i_{0})$, we have

[TABLE]

We also have $\widehat{R}-R=X+\partial Y$ for some $X$ and $Y$ with

[TABLE]

Consequently, we see that

[TABLE]

with

[TABLE]

That is, we have $\mathcal{F}(T^{\prime}_{i_{0}}-R)\leq\epsilon$.

Claim 3 has been proved.

Claim 4. For some $i$, the stopping conditions will be satisfied.

Proof of Claim. Applying Claim 2, we pass to a subsequence (without changing notation) for which the stopping condition (C2) holds for all $i$.

Arguing by contradiction, we suppose that

[TABLE]

holds for every $i$.

Since $T_{i}\in\mathcal{T}(i)$, there exists $S_{i}$ with $B=\partial S_{i}+\partial T_{i}$ and $\mathbb{M}(S_{i})<\epsilon_{i}$. Since $Q_{i}\in\mathcal{Q}(i)$, there exists $S^{\prime}_{i}$ with $\partial T^{\prime}_{i}=B_{i}=\partial S^{\prime}_{i}+\partial Q_{i}$ and $\mathbb{M}(S^{\prime}_{i})<\epsilon_{i}$.

Set

[TABLE]

We have

[TABLE]

and

[TABLE]

We may pass to a subsequence, again without changing notation, such that $P_{i}$ converges to $P^{*}$ and $S_{i}+T_{i}$ converges $T^{*}$. By the lower semicontinuity of mass and the right-hand inequality in (1), we see that both $P^{*}$ and $T^{*}$ are mass minimizers with boundary $B$. By construction, $P^{*}$ and $T^{*}$ are equal in $I(\epsilon_{1}/2)$. By the regularity theory of mass minimizers, the singular set of a minimizer cannot disconnect the surface. We have $P^{*}=T^{*}$.

The fact that $P^{*}=T^{*}$ tells us that $\mathcal{F}[(S_{i}+T_{i})-P_{i}]\rightarrow 0$, so we can write $(S_{i}+T_{i})-P_{i}=X_{i}+\partial Y_{i}$ with $\mathbb{M}(X_{i})+\mathbb{M}(Y_{i})\rightarrow 0$. Then applying the isoperimetric inequality to $X_{i}$, we see that we can write $(S_{i}+T_{i})-P_{i}=\partial Z_{i}$ with $\mathbb{M}(Z_{i})\rightarrow 0$.

On the other hand, observe that

[TABLE]

By the definition of $\mathcal{Q}(i)$, we have $T^{\prime}_{i}-Q_{i}-S^{\prime}_{i}=\partial W_{i}$ with $\mathbb{M}(W_{i})\geq\epsilon/2$. This last inequality contradicts $\mathbb{M}(Z_{i})\rightarrow 0$, because $W_{i}$ and $Z_{i}$ are $N$-dimensional integral currents in $\mathbb{R}^{N}$ having the same boundary, so in fact, they are equal.

Claim 4 has been proved.

Conclusion. Once the sequence $\epsilon_{i}$ satisfying the required conditions has been chosen, the algorithm proceeds as follows:

1. (A1)

Set $i=1$. 2. (A2)

Compute $T_{i}$. 3. (A3)

If the condition $\mathbb{M}[T_{i}\kern 1.99997pt\vrule width=0.59998pt,height=7.5347pt,depth=0.86108pt\vrule height=-0.34444pt,width=3.75pt,depth=0.86108pt\kern 1.99997pt\mathbb{R}^{N}\setminus\mathcal{K}]\leq\epsilon/2$ is satisfied, then advance to step (A4). Otherwise, increment $i$ and go to step (A2). 4. (A4)

Compute $Q_{i}$. 5. (A5)

If the condition $\mathbb{M}(Q_{i})\geq\mathbb{M}(T_{i}\kern 1.99997pt\vrule width=0.59998pt,height=7.5347pt,depth=0.86108pt\vrule height=-0.34444pt,width=3.75pt,depth=0.86108pt\kern 1.99997pt\mathcal{K})+3\,\epsilon_{i}$ is satisfied, then return $T^{\prime}_{i}$ and terminate the algorithm. Otherwise, increment $i$ and go to step (A2).

Claim 4 guarantees that the algorithm terminates after finitely many steps, while Claim 3 guarantees that the returned value $T^{\prime}_{i}$ is the desired approximation.