Plateau's Problem: What's Next

TL;DR

This paper reviews the evolution of Plateau's problem, a fundamental question in mathematics about finding minimal surfaces with boundary constraints, highlighting key developments, techniques, and open challenges over the past century.

Contribution

It synthesizes historical and recent advances in Plateau's problem, emphasizing new methods and unresolved issues in the study of minimal surfaces.

Findings

Key techniques from Federer, Fleming, Reifenberg, and Almgren

Progress on regularity and singularity analysis of minimal surfaces

Identification of open problems in the field

Abstract

Plateau's problem is not a single conjecture or theorem, but rather an abstract framework, encompassing a number of different problems in several related areas of mathematics. In its most general form, Plateau's problem is to find an element of a given collection \(\cal{C} \) of "surfaces" specified by some boundary constraint, which minimizes, or is a critical point of, a given "area" function \(F:\cal{C}\to \R \). In addition, one should also show that any such element satisfies some sort of regularity, that it be a sufficiently smooth manifold away from a well-behaved singular set. The choices apparent in making this question precise lead to a great many different versions of the problem. Plateau's problem has generated a large number of papers, inspired new fields of mathematics, and given rise to techniques which have proved useful in applications further afield. In this review we discuss a few highlights from the past hundred years, with special attention to papers of Federer, Fleming, Reifenberg and Almgren from the 1960's, and works by several groups, including ourselves, who have made significant progress on different aspects of the problem in recent years. A number of open problems are presented.