Rigidity of the gradient estimate for Einstein manifolds

TL;DR

This paper investigates conditions under which Ricci-flat manifolds with specific decay and volume growth properties are rigid, concluding they are flat or asymptotically locally Euclidean (ALE) under certain Green function bounds.

Contribution

It establishes new rigidity results for Ricci-flat manifolds based on Green function gradient bounds and curvature decay, extending previous classifications.

Findings

Manifolds are flat if Green function gradient is bounded below.

Curvature belongs to L^p for p ≥ 2 under quadratic decay and Euclidean volume growth.

Manifolds are proven to be ALE of optimal order under these conditions.

Abstract

We study the rigidity of Ricci-flat manifolds with quadratic curvature decay under conditions on the Green function. We show that if the gradient of the Green function is uniformly bounded from below, then the manifold is flat. Furthermore, we prove that for a Ricci-flat manifold with quadratic curvature decay and Euclidean volume growth, the curvature is in $L^p$ for any $p \ge 2$. Combining with Cheeger-Tian \cite{CT} and Kr\"oncke-Szab\'o \cite{KS}, we obtain that the manifold must be ALE of optimal order.