Noncollapsing estimate for the Ricci-Bourguignon flow

Abstract

The Ricci-Bourguignon flow (R-B flow) is a general geometric evolving equation, which includes or relates to some famous geometric flows, for example the Ricci flow and the Yamabe flow, etc. In this paper we shall prove that for the R-B flow evolving on [0,T), whose first eigenvalue λ0 of the operator −△+(1−(n−1)ρ)24(1−2(n−1)ρ)R for the initial metric g(0) is positive, or T>0 is finite, an upper bound assumption of the scalar curvature implies a noncollapsing estimate of the volume, uniformly for all time. In order to derive this noncollapsing estimate, we firstly establish a logarithmic Sobolev inequality along the R-B flow, by using the monotone formula for the Perelman's functional, and then we can derive a Sobolev inequality along the R-B flow.