Hamilton's Theorem (on the compactness of pinched hypersurfaces) via mean curvature flow

TL;DR

This paper rigorously applies mean curvature flow to prove Hamilton's theorem on the compactness of properly embedded hypersurfaces with pinched, bounded curvature, providing a new geometric analysis approach.

Contribution

It introduces a rigorous method using mean curvature flow to establish Hamilton's compactness theorem for hypersurfaces with curvature bounds.

Findings

Mean curvature flow effectively proves hypersurface compactness.

Pinched curvature conditions lead to compactness results.

New geometric analysis techniques are developed for hypersurfaces.

Abstract

We make rigorous an old idea of using mean curvature flow to prove a theorem of Richard Hamilton on the compactness of proper hypersurfaces with pinched, bounded curvature.