The nonuniqueness of the tangent cones at infinity of Ricci-flat manifolds

Abstract

Colding and Minicozzi established the uniqueness of the tangent cones at infinity of Ricci-flat manifolds with Euclidean volume growth where at least one tangent cone at infinity has a smooth cross section. In this paper, we raise an example of a Ricci-flat manifold implying that the assumption for the volume growth in the above result is essential. More precisely, we construct a complete Ricci-flat manifold of dimension $4$ with non-Euclidean volume growth that has infinitely many tangent cones at infinity where one of them has a smooth cross section.