The Maximal Graph Dirichlet Problem in Semi-Euclidean Spaces

TL;DR

This paper proves the existence of spacelike maximal graphs with zero mean curvature in semi-Euclidean spaces, extending results to higher codimensions under specific boundary conditions.

Contribution

It establishes the existence of solutions to the maximal graph Dirichlet problem in semi-Euclidean spaces for higher codimension graphs, which was previously unresolved.

Findings

Existence of solutions under certain boundary conditions

Results applicable to graphs of codimension greater than 1

Extension of maximal graph theory to semi-Euclidean spaces

Abstract

The maximal graph Dirichlet problem asks whether there exists a spacelike graph, in a semi-Euclidean space, with a given boundary and with mean curvature everywhere zero. We prove the existence of solutions to this problem under certain assumptions on the given boundary. Most importantly, the results proved here will hold for graphs of codimension greater than 1.