How Riemannian Manifolds Converge: A Survey

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.

Solicited survey article for a volume of articles in honor of Cheeger's 65th Birthday. Version 2 adds additional references concerning convergence of Lorentzian manifolds and updates citations

References in corpus (6)

• Currents and Flat Chains Associated to Varifolds, with an Application to Mean Curvature Flow
• A Lorentzian Gromov-Hausdoff notion of distance
• The moduli space of isometry classes of globally hyperbolic spacetimes
• The limit space of a Cauchy sequence of globally hyperbolic spacetimes
• The Topology of Open Manifolds with Nonnegative Ricci Curvature
• The Intrinsic Flat Distance between Riemannian Manifolds and other Integral Current Spaces

Cited by in corpus (3)

• Cheeger-Gromov compactness for manifolds with boundary
• Evolutionary de Rham-Hodge method
• Boundary harmonic coordinates on manifolds with boundary in low regularity