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

TL;DR

This paper constructs a 4-dimensional Ricci-flat manifold with non-Euclidean volume growth that has multiple tangent cones at infinity, demonstrating the necessity of volume growth assumptions in tangent cone uniqueness results.

Contribution

It provides a counterexample showing that Ricci-flat manifolds can have non-unique tangent cones at infinity when volume growth conditions are not met.

Findings

Existence of a 4D Ricci-flat manifold with multiple tangent cones at infinity.

One tangent cone has a smooth cross section, the other does not.

Volume growth condition is essential for tangent cone uniqueness.

Abstract

It is shown by Colding and Minicozzi the uniqueness of the tangent cone at infinity of Ricci-flat manifolds with Euclidean volume growth which has at least one tangent cone at infinity with a smooth cross section. In this article we raise an example of the 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 who has at least two distinct tangent cones at infinity and one of them has a smooth cross section.