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

Hattori, Kota

doi:10.2140/gt.2017.21.2683

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

Kota Hattori

Geometry & Topology 21 (2017) 2683–2723

DOI: 10.2140/gt.2017.21.2683

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.

Keywords

hyper-Kähler, tangent cone at infinity, Ricci flat manifold

Mathematical Subject Classification 2010

Primary: 53C23

References

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

Volume 21, issue 5 (2017)