Rigidity of the gradient estimate for Einstein manifolds

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.