On uniqueness of tangent cones for Einstein manifolds

We show that for any Ricci-flat manifold with Euclidean volume growth the tangent cone at infinity is unique if one tangent cone has a smooth cross-section. Similarly, for any noncollapsing limit of Einstein manifolds with uniformly bounded Einstein constants, we show that local tangent cones are unique if one tangent cone has a smooth cross-section.National Science Foundation (U.S.) (NSF Grant DMS 0906233)National Science Foundation (U.S.) (NSF Grant DMS 11040934)National Science Foundation (U.S.) (NSF FRG grant DMS 0853501)National Science Foundation (U.S.) (NSF FRG grant DMS 0854774