Scalar Curvature and Intrinsic Flat Convergence

TL;DR

This paper surveys the concept of intrinsic flat convergence for sequences of Riemannian manifolds with scalar curvature bounds, highlighting its advantages over traditional convergence notions and discussing open problems and applications in geometry and physics.

Contribution

It provides a comprehensive review of intrinsic flat convergence, its properties, compactness theorems, and explores its applications and open problems in geometric analysis and general relativity.

Findings

Intrinsic flat convergence can handle sequences not converging smoothly or Gromov-Hausdorff.

Applications in General Relativity demonstrate the usefulness of intrinsic flat limits.

Open problems suggest broader applicability in geometric analysis.

Abstract

Herein we present open problems and survey examples and theorems concerning sequences of Riemannian manifolds with uniform lower bounds on scalar curvature and their limit spaces. Examples of Gromov and of Ilmanen which naturally ought to have certain limit spaces do not converge with respect to smooth or Gromov-Hausdorff convergence. Thus we focus here on the notion of Intrinsic Flat convergence, developed jointly with Wenger. This notion has been applied successfully to study sequences that arise in General Relativity. Gromov has suggested it should be applied in other settings as well. We first review intrinsic flat convergence, its properties, and its compactness theorems, before presenting the applications and the open problems.