The Penrose inequality in general relativity and volume comparison theorems involving scalar curvature (thesis)

TL;DR

This thesis demonstrates how minimal surface techniques can prove the Penrose inequality and establish a new volume comparison theorem involving scalar curvature for 3-manifolds in general relativity.

Contribution

It introduces a novel application of minimal surface methods to prove the Penrose inequality and derive a volume comparison theorem related to scalar curvature.

Findings

Proved the Penrose inequality for specific classes of 3-manifolds.

Established a new volume comparison theorem involving scalar curvature.

Showed the effectiveness of minimal surface techniques in geometric inequalities.

Abstract

In this thesis we describe how minimal surface techniques can be used to prove the Penrose inequality in general relativity for two classes of 3-manifolds. We also describe how a new volume comparison theorem involving scalar curvature for 3-manifolds follows from these same techniques.