Non-Kähler expanding Ricci solitons, Einstein metrics, and exotic cone structures

Maria Buzano, Andrew S. Dancer, Michael Gallaugher and McKenzie Wang

Vol. 273 (2015), No. 2, 369–394

DOI: 10.2140/pjm.2015.273.369

Abstract

We consider complete multiple warped product type Riemannian metrics on manifolds of the form ${ℝ}^2\times M_2\times \cdots \times M_r$, where $r\ge 2$ and $M_i$ are arbitrary closed Einstein spaces with positive scalar curvature. We construct on these spaces a family of non-Kähler, non-Einstein, expanding gradient Ricci solitons with conical asymptotics as well as a family of Einstein metrics with negative scalar curvature. The $2$-dimensional Euclidean space factor allows us to obtain homeomorphic but not diffeomorphic examples which have analogous cone structure behaviour at infinity. We also produce numerical evidence for complete expanding solitons on the vector bundles whose sphere bundles are the twistor or $Sp\left(1\right)$ bundles over quaternionic projective space.

Keywords

expanders, gradient Ricci solitons, Einstein metrics, exotic structures

Mathematical Subject Classification 2010

Primary: 53C25

Secondary: 53C44

Milestones

Received: 14 February 2014

Accepted: 6 May 2014

Published: 23 December 2014