Smooth metric measure spaces and quasi-einstein metrics

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Abstract

Smooth metric measure spaces have been studied from the two different perspectives of Bakry-Émery and Chang-Gursky-Yang, both of which are closely related to work of Perelman on the Ricci flow. These perspectives include a generalization of the Ricci curvature and the associated quasi-Einstein metrics, which include Einstein metrics, conformally Einstein metrics, gradient Ricci solitons and static metrics. In this paper, we describe a natural perspective on smooth metric measure spaces from the point of view of conformal geometry and show how it unites these earlier perspectives within a unified framework. We offer many results and interpretations which illustrate the unifying nature of this perspective, including a natural variational characterization of quasi-Einstein metrics as well as some interesting families of examples of such metrics.

Original languageEnglish (US)
Article number1250110
JournalInternational Journal of Mathematics
Volume23
Issue number10
DOIs
StatePublished - Oct 1 2012

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All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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