The Perron Method and the Non-Linear Plateau Problem

TL;DR

This paper introduces a new technique based on the Perron method for solving the Plateau problem for constant curvature hypersurfaces, demonstrating its effectiveness through an existence theorem for hypersurfaces with constant Gaussian curvature in Euclidean space.

Contribution

It presents a novel Perron method approach for the non-linear Plateau problem, extending the theory to hypersurfaces of constant Gaussian curvature.

Findings

Existence of hypersurfaces with constant Gaussian curvature in Euclidean space.

Application of the Perron method to non-linear geometric problems.

Abstract

We describe a novel technique for solving the Plateau problem for constant curvature hypersurfaces based on recent work of Harvey and Lawson. This is illustrated by an existence theorem for hypersurfaces of constant Gaussian curvature in $\Bbb{R}^{n+1}$.