A Gaussian upper bound of the conjugate heat equation along Ricci-harmonic flow

Xian-Gao Liu and Kui Wang

Vol. 287 (2017), No. 2, 465–484

DOI: 10.2140/pjm.2017.287.465

Abstract

We mainly study the Ricci-harmonic flow. Using the monotonicity formulae of entropies, we show a uniform Sobolev inequality along Ricci-harmonic flow. Furthermore, we obtain a Gaussian upper bound for the fundamental solutions of the conjugate heat equation via Moser iteration and Sobolev inequality.

Keywords

Ricci-harmonic flow, Sobolev inequality, Gaussian upper bound

Mathematical Subject Classification 2010

Primary: 35B40, 53C44, 35K05

Milestones

Received: 21 August 2015

Revised: 8 June 2016

Accepted: 22 September 2016

Published: 9 March 2017