Differential Harnack Estimates for Backward Heat Equations with Potentials under the Ricci Flow

By Xiaodong Cao. 2008

71 citations recorded by the source corpus.

In this paper, we derive a general evolution formula for possible Harnack quantities. As a consequence, we prove several differential Harnack inequalities for positive solutions of backward heat-type equations with potentials (including the conjugate heat equation) under the Ricci flow. We shall also derive Perelman's Harnack inequality for the fundamental solution of the conjugate heat equation under the Ricci flow.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/23f143aa04b1e2cf78033b70476e2612adbfe0b7

How researchers see this paper

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In Applied Mathematics

1. Xiaodong Cao
2. Qi S. Zhang
3. Mihai Băileşteanu
4. Artem Pulemotov
5. R. Hamilton
6. Benjamin Fayyazuddin Ljungberg
7. Bowei Liu
8. Bing Wang

In Geometry and Topology

1. Qi S. Zhang
2. R. Hamilton
3. Xiaodong Cao
4. Mihai Băileşteanu
5. Artem Pulemotov
6. Chi Li
7. Ben Andrews
8. A. Naber

Citations in the corpus

• Differential Harnack Estimates for Time-Dependent Heat Equations with Potentials (2008, methodology): “More importantly, the method used here will help us searching for more interesting Harnack inequalities in the Ricci flow as well as in other geometric flows (for example, see [Cao08]).”
• Differential Harnack estimates for a nonlinear heat equation (2013, background): “In [1,2], the first author used a new idea of undetermined coefficients to systematically search for new Harnack inequalities.”
• 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].”
• Gradient estimates for the heat equation under the Ricci flow (2009, background): “Hamilton’s result were proposed in [31, 36, 5, 6].”
• 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.”