Embedded Plateau Problem

TL;DR

This paper proves the existence of an area-minimizing, smooth, embedded disk bounded by a simple closed curve in certain 3-manifolds, extending classical Plateau problem results to more general settings.

Contribution

It establishes the existence and regularity of area-minimizing embedded disks bounded by a given curve in closed and convex 3-manifolds, generalizing previous results.

Findings

Existence of area-minimizing embedded disks in closed 3-manifolds.

Smoothness and minimality of the disk except at boundary intersections.

Extension of results to manifolds with convex boundary.

Abstract

We show that if C is a simple closed curve bounding an embedded disk in a closed 3-manifold M, then there exists a disk D in M with boundary C such that D minimizes the area among the embedded disks with boundary C. Moreover, D is smooth, minimal and embedded everywhere except where the boundary C meets the interior of D. The same result is also valid for homogenously regular manifolds with sufficiently convex boundary.