Scalar curvature lower bound under integral convergence

TL;DR

This paper proves that sequences of smooth metrics with almost non-negative scalar curvature in an integral sense converge to a metric with a pointwise scalar curvature lower bound, bridging integral and pointwise curvature conditions.

Contribution

It establishes a link between integral scalar curvature bounds of converging metrics and pointwise scalar curvature bounds of the limit metric.

Findings

Sequences with integral scalar curvature bounds converge to a metric with pointwise bounds.

The result applies to $C^2$ metrics converging in $C^0$ sense.

Provides a method to infer pointwise curvature bounds from integral conditions.

Abstract

In this work, we consider sequences of $C^2$ metrics which converge to a $C^2$ metric in $C^0$ sense. We show that if the scalar curvature of the sequence is almost non-negative in the integral sense, then the limiting metric has scalar curvature lower bound in point-wise sense.