Existence of least-area discs

Abstract

We prove that every closed Lipschitz curve in Euclidean space extends to a Lipschitz disc of least area, with Lipschitz constant controlled by that of the boundary curve. Our proof gives a new approach to the classical Douglas--Radó solution of the Plateau problem but accommodates arbitrary self-intersections and avoids Sobolev maps. The argument extends to metric spaces with curvature bounded above in the sense of Alexandrov.

Proof found without AI