The non-linear Plateau problem in non-positively curved manifolds

TL;DR

This paper proves the existence of hypersurfaces with specific curvature properties in non-positively curved manifolds using the Perron method, advancing geometric analysis in differential geometry.

Contribution

It introduces a new application of the Perron method to establish hypersurfaces with prescribed special Lagrangian curvature in non-positive curvature settings.

Findings

Existence of hypersurfaces with prescribed special Lagrangian curvature.

Application of Perron method in non-positively curved manifolds.

Extension of geometric analysis techniques to complex curvature conditions.

Abstract

Using the Perron method, we prove the existence of hypersurfaces of prescribed special Lagrangian curvature with prescribed boundary inside complete Riemannian manifolds of non-positive curvature.