Geometric relations of stable minimal surfaces and applications

TL;DR

This paper derives geometric relations for stable minimal surfaces in three-manifolds with scalar curvature bounds, leading to new insights on isoperimetric properties and collapse phenomena.

Contribution

It generalizes a formula to establish geometric relations on stable minimal surfaces and applies these to analyze isoperimetric collapse in curved three-manifolds.

Findings

Non-local rigidity results for isoperimetric properties

Relations between local and global isoperimetric behaviors

Insights into isoperimetric collapse phenomena

Abstract

We establish some a priori geometric relations on stable minimal surfaces lying inside three-manifolds with scalar curvature uniformly bounded below. The relations are based on a slight generalization of a formula due to Castillon. We apply it to prove non-local rigidity results in the particular sense that they express how local isoperimetric properties in some region affect the local isoperimetric properties in any other region. We present applications to understand the notion of isoperimetric collapse on three-manifolds with scalar curvature uniformly bounded below.