A survey on the Convergence of Manifolds with Boundary

TL;DR

This survey comprehensively reviews existing theorems and introduces new results on the convergence properties of classes of Riemannian manifolds with boundary, emphasizing precompactness conditions.

Contribution

It consolidates prior work and presents novel findings on the convergence and precompactness of Riemannian manifolds with boundary.

Findings

Summary of key precompactness theorems by Kodani, Anderson et al., Wong

Introduction of new convergence results by Knox and the author with Sormani

Enhanced understanding of boundary effects on manifold convergence

Abstract

This survey reviews precompactness theorems for classes of Riemannian manifolds with boundary. We begin with the works of Kodani, Anderson-Katsuda-Kurylev-Lassas-Taylor and Wong. We then present new results of Knox and the author with Sormani.