Stability of Euclidean space under Ricci flow

TL;DR

This paper proves that small perturbations of the Euclidean metric on R^n under Ricci flow remain stable and converge back to the Euclidean metric over time, using a new monotone integral quantity.

Contribution

It introduces a stability result for Ricci flow near Euclidean space and a new monotone integral to measure metric deviation.

Findings

Ricci harmonic map heat flow exists globally for asymptotically Euclidean metrics

Flow converges uniformly to Euclidean metric as time approaches infinity

A new monotone integral quantity effectively measures deviation from Euclidean space

Abstract

We study the Ricci flow for initial metrics which are C^0 small perturbations of the Euclidean metric on R^n. In the case that this metric is asymptotically Euclidean, we show that a Ricci harmonic map heat flow exists for all times, and converges uniformly to the Euclidean metric as time approaches infinity. In proving this stability result, we introduce a monotone integral quantity which measures the deviation of the evolving metric from the Euclidean metric. We also investigate the convergence of the diffeomorphisms relating Ricci harmonic map heat flow to Ricci flow.