Compactness for immersions of prescribed Gaussian curvature II - geometric aspects

TL;DR

This paper develops compactness results and degree theory for locally convex immersions with prescribed Gaussian curvature, enabling solutions to the Plateau problem in Hadamard manifolds.

Contribution

It introduces new compactness and degree theory tools for immersions with prescribed Gaussian curvature, extending the understanding of geometric boundary value problems.

Findings

Established a boundary compactness result for locally convex immersions.

Developed a mod 2 degree theory for constant and prescribed Gaussian curvature immersions.

Solved the Plateau problem for such immersions in Hadamard manifolds.

Abstract

We develop a compactness result near the boundary for families of locally convex immersions. We also develop a mod 2 degree theory for immersion of constant (and prescribed) Gaussian curvature with prescribed boundary. These are then used to solve the Plateau problem for immersions of constant (and prescribed) Gaussian curvature in general Hadamard manifolds.