#### Volume 21, issue 5 (2017)

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

### Kota Hattori

Geometry & Topology 21 (2017) 2683–2723
##### 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.

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##### Keywords
hyper-Kähler, tangent cone at infinity, Ricci flat manifold
Primary: 53C23
##### Publication
Received: 10 May 2015
Revised: 12 October 2016
Accepted: 13 October 2016
Published: 15 August 2017
Proposed: Tobias H. Colding
Seconded: Dmitri Burago, Bruce Kleiner
##### Authors
 Kota Hattori Department of Mathematics Keio University 3-14-1 Hiyoshi Kohoku Yokohama 223-8522 Japan