### Regularity of the Singular Set of Two-Dimensional Area-Minimizing Flat
Chains Modulo 3 in R^{3}

Inventiones Mathematicae, Vol. 22 (1973), 119-159

Introduction

Determining the existence and structure of surfaces of minimum area having a given boundary is a problem of long-standing interest. The success of geometric measure theoretic methods in recent years in showing the existence and regularity almost everywhere of solutions to a variety of different formulations of the problem of least area (the class of problems frequently collectively called Plateau's Problem) has now focused increased attention on the problem of determining the structure of the non-regular, or singular, points of these solutions. The results of this thesis are a contribution to the study of the lower dimensional regularity of certain of these singular sets.

The class of surfaces we consider here, namely rectifiable flat chains modulo 3 whose boundaries are also rectifiable flat chains modulo 3, can be defined in a number of ways. They are characterized by the property that both they and their boundaries simultaneously agree, except in a set of arbitrarily small two and one-dimensional area respectively, with the images of C^{1} singular chains with coefficients in the integers mod 3, the particular chains depending on the degree of approximation desired. One dimensional singularities can arise in these surfaces because of cancellation modulo 3. For instance, consider the surface Y consisting of three half disks with common diameter and at angles of 120° to each other (in spherical coordinates

Y = {(r, θ, φ): r ≤ 1, θ=0 or 2π/3 or 4π/3}).

This Y is the support of a rectifiable flat chain modulo 3 with boundary support consisting of three half circles; indeed, Y (suitably oriented) is the area minimizing flat chain modulo 3 having this boundary (correspondingly oriented) and it represents the soap film which physically forms on a wire frame consisting of three such half circles.

**Table of Contents**

- Introduction
- Preliminaries
- The Spherical Epiperimetric Inequality
- The Epiperimetric Inequality for Y-Surfaces
- Uniqueness of Tangent Cones and Smoothness of Singularities