The logarithmic Sobolev inequality along the Ricci flow

TL;DR

This paper establishes a time-uniform logarithmic Sobolev inequality along the Ricci flow, depending only on initial geometric data, leading to uniform Sobolev inequalities and noncollapsing estimates that extend to flows with surgeries.

Contribution

It derives a new logarithmic Sobolev inequality valid for all times along the Ricci flow, without restrictions, based solely on initial conditions and eigenvalue positivity.

Findings

Uniform Sobolev inequality along Ricci flow

Time-uniform kappa-noncollapsing estimate

Extension of results to Ricci flow with surgeries

Abstract

We derive a logarithmic Sobolev inequality along the Ricci flow without any restriction on time, which depends only on the initial metric via rudimentary geometric data, assuming only that a certain first eigenvalue is positive. As a consequence we obtain a uniform Sobolev inequality along the Ricci flow without any restriction on time. One application of it is a uniform kappa-noncollapsing estimate which holds true for all time. We also obtain similar results for bounded time without assuming the eigenvalue condition. The results extend to the Ricci flow with surgeries.