Uniqueness of Warped Product Einstein Metrics and Applications

Chenxu He, Peter Petersen, William Wylie

Research output: Contribution to journalArticle

1 Scopus citations

Abstract

We prove that complete warped product Einstein metrics with isometric bases, simply connected space form fibers, and the same Ricci curvature and dimension are isometric. In the compact case we also prove that the warping functions must be the same up to scaling, while in the non-compact case there are simple examples showing that the warping function is not unique. These results follow from a structure theorem for warped product Einstein spaces which is proven by applying the results in our earlier paper He et al. (Asian J Math 2011) to a vector space of virtual Einstein warping functions. We also use the structure theorem to study gap phenomena for the dimension of the space of warping functions and the isometry group of a warped product Einstein metric.

Original languageEnglish (US)
Pages (from-to)2617-2644
Number of pages28
JournalJournal of Geometric Analysis
Volume25
Issue number4
DOIs
StatePublished - Oct 1 2015

Keywords

  • Einstein manifold
  • Uniqueness
  • Warped product

ASJC Scopus subject areas

  • Geometry and Topology

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