Communications in Analysis and Geometry

Volume 17 (2009)

Number 4

Sharp logarithmic Sobolev inequalities on gradient solitons and applications

Pages: 721 – 753

DOI: https://dx.doi.org/10.4310/CAG.2009.v17.n4.a7

Authors

José A. Carrillo (ICREA and Departament de Matemàtiques, Universitat Autònoma de Barcelona, Spain)

Nei Ni (Department of Mathematics, University of California at San Diego)

Abstract

We show that gradient shrinking, expanding or steady Riccisolitons have potentials leading to suitable reference probabilitymeasures on the manifold. For shrinking solitons, as well asexpanding solitons with nonnegative Ricci curvature, thesereference measures satisfy sharp logarithmic Sobolev inequalitieswith lower bounds characterized by the geometry of the manifold.The geometric invariant appearing in the sharp lower bound isshown to be nonnegative. We also characterize the expanders whensuch invariant is zero. In the proof, various useful volumegrowth estimates are also established for gradient shrinking andexpanding solitons. In particular, we prove that the {\itasymptotic volume ratio} of any gradient shrinking soliton withnonnegative Ricci curvature must be zero.

Published 1 January 2009