Sharp logarithmic Sobolev inequalities on gradient solitons and applications

José A. Carrillo, Lei Ni

Research output: Contribution to journalArticleResearchpeer-review

37 Citations (Scopus)

Abstract

We show that gradient shrinking, expanding or steady Ricci solitons have potentials leading to suitable reference probability measures on the manifold. For shrinking solitons, as well as expanding solitons with nonnegative Ricci curvature, these reference measures satisfy sharp logarithmic Sobolev inequalities with lower bounds characterized by the geometry of the manifold. The geometric invariant appearing in the sharp lower bound is shown to be nonnegative. We also characterize the expanders when such invariant is zero. In the proof, various useful volume growth estimates are also established for gradient shrinking and expanding solitons. In particular, we prove that the asymptotic volume ratio of any gradient shrinking soliton with nonnegative Ricci curvature must be zero.
Original languageEnglish
Pages (from-to)721-753
JournalCommunications in Analysis and Geometry
Volume17
Issue number4
DOIs
Publication statusPublished - 1 Jan 2009

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