The Energy of a Smooth Metric Measure Space and Applications

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Abstract

We introduce and study the notion of the energy of a smooth metric measure space, which includes as special cases the Yamabe constant and Perelman’s ν-entropy. We then investigate some properties the energy shares with these constants, in particular its relationship with the κ-noncollapsing property. Finally, we use the energy to prove a precompactness theorem for the space of compact quasi-Einstein smooth metric measure spaces, in the spirit of similar results for Einstein metrics and gradient Ricci solitons.

Original languageEnglish (US)
Pages (from-to)616-667
Number of pages52
JournalJournal of Geometric Analysis
Volume25
Issue number1
DOIs
StatePublished - Jan 1 2013

All Science Journal Classification (ASJC) codes

  • Geometry and Topology

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