How Riemannian Manifolds Converge: A Survey

TL;DR

This survey provides an overview of various extrinsic and intrinsic notions of convergence for manifolds, illustrating key concepts with examples and discussing potential future directions in the field.

Contribution

It compiles and explains different convergence notions for manifolds, highlighting their properties and applications, and explores possible new directions like intrinsic varifold convergence.

Findings

Different notions of convergence are suited for various geometric and topological analyses.

Intrinsic and extrinsic convergence concepts have distinct properties and applications.

The survey suggests future research directions in convergence of Lorentzian manifolds and area convergence.

Abstract

This is an intuitive survey of extrinsic and intrinsic notions of convergence of manifolds complete with pictures of key examples and a discussion of the properties associated with each notion. We begin with a description of three extrinsic notions which have been applied to study sequences of submanifolds in Euclidean space: Hausdorff convergence of sets, flat convergence of integral currents, and weak convergence of varifolds. We next describe a variety of intrinsic notions of convergence which have been applied to study sequences of compact Riemannian manifolds: Gromov-Hausdorff convergence of metric spaces, convergence of metric measure spaces, Instrinsic Flat convergence of integral current spaces, and ultralimits of metric spaces. We close with a speculative section addressing possible notions of intrinsic varifold convergence, convergence of Lorentzian manifolds and area convergence.