A direct approach to Plateau's problem

TL;DR

This paper introduces a new compactness principle based on Radon measures for solving Plateau's problem, providing alternative proofs and addressing open questions in geometric measure theory.

Contribution

It presents a novel compactness principle applicable to various formulations of Plateau's problem, using elementary comparison arguments and measure theory techniques.

Findings

New compactness principle for Plateau's problem

Alternative proof of Harrison and Pugh's theorem

Answer to Guy David's open question

Abstract

We provide a compactness principle which is applicable to different formulations of Plateau's problem in codimension one and which is exclusively based on the theory of Radon measures and elementary comparison arguments. Exploiting some additional techniques in geometric measure theory, we can use this principle to give a different proof of a theorem by Harrison and Pugh and to answer a question raised by Guy David.