A gradient estimate for all positive solutions of the conjugate heat equation under Ricci flow

By Shi-jiang Kuang, Qi S. Zhang. 2006

78 citations recorded by the source corpus.

Abstract We establish a point-wise gradient estimate for all positive solutions of the conjugate heat equation. This contrasts to Perelman's point-wise gradient estimate which works mainly for the fundamental solution rather than all solutions. Like Perelman's estimate, the most general form of our gradient estimate does not require any curvature assumption. Moreover, assuming only lower bound on the Ricci curvature, we also prove a localized gradient estimate similar to the Li–Yau estimate for the linear Schrodinger heat equation. The main difference with the linear case is that no assumptions on the derivatives of the potential (scalar curvature) are needed. A classical Harnack inequality follows.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ead972ca3c5e49a6e309f381b33308dd31e3616f

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

The conjugate heat equation and Ancient solutions of the Ricci flow (2010, methodology): “Perelman successfully proved a non-collapsing result using the conjugate heat equation associated with the Ricci flow, the authors ([3] and [16]) proved a differential Harnack inequality for positive solutions of (1.”
Differential Harnack Estimates for Time-Dependent Heat Equations with Potentials (2008, result): “Such Harnack inequality for the conjugate heat equation has also been proved by S. Kuang and Q. Zhang [ KZ06 ] recently.”
Gradient estimates for positive solutions of the heat equation under geometric flow (2011, unknown): “Since then, there have been many works on gradient estimates along the Ricci flow or the conjugate Ricci flow for the solution of the heat equation or the conjugate heat equation; examples include [Cao 2008; Cao and Hamilton 2009; Kuang and Zhang 2008; Zhang 2006].”
Pseudolocality for the Ricci Flow and Applications (2007, methodology): “As an application of the LYH inequality, Perelman proved a pseudolocality result for the Received by the editors July 3, 2008.”