Sharp logarithmic Sobolev inequalities on gradient solitons and applications

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 soltions 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 {\it asymptotic volume ratio} of any gradient shrinking soliton with nonnegative Ricci curvature must be zero.

submitted

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Cited by in corpus (6)

• A compactness theorem for complete Ricci shrinkers
• Non-Kähler Expanding Ricci Solitons II
• Rigidity of asymptotically conical shrinking gradient Ricci solitons
• Liouville theorems, Volume growth, and volume comparison for Ricci shrinkers
• Time analyticity for heat equation on gradient shrinking Ricci solitons
• Non-singular solutions of normalized Ricci flow on noncompact manifolds of finite volume