The Non-Liner Dirichlet Problem in Hadamard Manifolds

TL;DR

This paper proves existence theorems for hypersurfaces with constant special Lagrangian curvature in Hadamard manifolds, employing continuity and Perron methods for refined results.

Contribution

It introduces new existence results for the Dirichlet problem in Hadamard manifolds using novel combination of continuity and Perron methods.

Findings

Existence theorems for hypersurfaces with constant special Lagrangian curvature.

A-priori estimates valid in any ambient manifold.

Refined solutions via iterative Perron method.

Abstract

We proof existence theorems for the Dirichlet problem for hypersurfaces of constant special Lagrangian curvature in Hadamard manifolds. The first results are obtained using the continuity method and approximation and then refined using two iterations of the Perron method. The a-priori estimates used in the continuity method are valid in any ambient manifold.