Uniqueness of warped product Einstein metrics and applications
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 "Warped product rigidity" 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.
Keywords
Cite
@article{arxiv.1110.2456,
title = {Uniqueness of warped product Einstein metrics and applications},
author = {Chenxu He and Peter Petersen and William Wylie},
journal= {arXiv preprint arXiv:1110.2456},
year = {2013}
}
Comments
23 pages. New applications to the uniqueness of warped product Einstein metrics have been added and the exposition has been revised. The results on the Lie groups have been extended to general homogeneous spaces and moved to the preprint "Warped product Einstein metrics on homogeneous spaces and homogeneous Ricci solitons", see arXiv: 1302.0246v1