Defining Pointwise Lower Scalar Curvature Bounds for $C^0$ Metrics with Regularization by Ricci Flow

TL;DR

This paper reviews recent methods using Ricci flow to define weak lower scalar curvature bounds for continuous metrics, exploring their properties, applications, and relation to other curvature notions.

Contribution

It introduces a new class of local definitions for scalar curvature bounds applicable to $C^0$ metrics and analyzes their properties and implications.

Findings

Defines weak lower scalar curvature bounds for $C^0$ metrics using Ricci flow.

Establishes properties and applications of these definitions in metric convergence.

Explores relationships with other generalized scalar curvature notions.

Abstract

We survey some recent work using Ricci flow to create a class of local definitions of weak lower scalar curvature bounds that is well defined for $C^0$ metrics. We discuss several properties of these definitions and explain some applications of this approach to questions regarding uniform convergence of metrics with scalar curvature bounded below. Finally, we consider the relationship between this approach and some other generalized notions of lower scalar curvature bounds.