On the Perelman's reduced entropy and Ricci flat manifolds with maximal volume growth

TL;DR

This paper investigates Ricci flat manifolds with maximal volume growth, demonstrating quadratic curvature decay under certain conditions using Perelman's reduced volume, and explores related applications.

Contribution

It establishes quadratic curvature decay for Ricci flat manifolds with maximal volume growth and vanishing curvature at infinity, using Perelman's reduced volume.

Findings

Ricci flat manifolds with maximal volume growth have quadratic curvature decay.

If curvature tends to zero at infinity, then quadratic decay is guaranteed.

Applications include further geometric properties of such manifolds.

Abstract

In this paper, we study the Ricci flat manifolds with maximal volume growth using Perelman's reduced volume of Ricci flow. We show that if $(M^n,g)$ is an noncompact complete Ricci flat manifold with maximal volume growth satisfying $|Rm|(x)\to 0$ as $d(x)=d_g(x,p)\to \infty$, then $M^n$ has the quadratic curvature decay. Some applications to this result are also presented.