### 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.

Language | English (US) |
---|---|

Pages | 2617-2644 |

Number of pages | 28 |

Journal | Journal of Geometric Analysis |

Volume | 25 |

Issue number | 4 |

DOIs | |

State | Published - Oct 1 2015 |

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### Keywords

- Einstein manifold
- Uniqueness
- Warped product

### ASJC Scopus subject areas

- Geometry and Topology

### Cite this

*Journal of Geometric Analysis*,

*25*(4), 2617-2644. DOI: 10.1007/s12220-014-9528-8

**Uniqueness of Warped Product Einstein Metrics and Applications.** / He, Chenxu; Petersen, Peter; Wylie, William.

Research output: Research - peer-review › Article

*Journal of Geometric Analysis*, vol 25, no. 4, pp. 2617-2644. DOI: 10.1007/s12220-014-9528-8

}

TY - JOUR

T1 - Uniqueness of Warped Product Einstein Metrics and Applications

AU - He,Chenxu

AU - Petersen,Peter

AU - Wylie,William

PY - 2015/10/1

Y1 - 2015/10/1

N2 - 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.

AB - 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.

KW - Einstein manifold

KW - Uniqueness

KW - Warped product

UR - http://www.scopus.com/inward/record.url?scp=85027918485&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85027918485&partnerID=8YFLogxK

U2 - 10.1007/s12220-014-9528-8

DO - 10.1007/s12220-014-9528-8

M3 - Article

VL - 25

SP - 2617

EP - 2644

JO - Journal of Geometric Analysis

T2 - Journal of Geometric Analysis

JF - Journal of Geometric Analysis

SN - 1050-6926

IS - 4

ER -