The Logarithmic Sobolev and Sobolev Inequalities Along the Ricci Flow

Abstract

Based on Perelman’s entropy monotonicity, uniform logarithmic Sobolev inequalities along the Ricci flow are derived. Then uniform Sobolev inequalities along the Ricci flow are derived via harmonic analysis of the integral transform of the relevant heat operator. These inequalities are fundamental analytic properties of the Ricci flow. They are also extended to the volume-normalized Ricci flow and the Kähler–Ricci flow.

Keywords

Sobolev inequality / Logarithmic Sobolev inequality / Ricci flow / Heat operator