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arXiv · 1006.0540

The Conjugate Heat Equation and Ancient Solutions of the Ricci Flow

Xiaodong Cao·Qi S. Zhang
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Abstract

We prove Gaussian type bounds for the fundamental solution of the conjugate heat equation evolving under the Ricci flow. As a consequence, for dimension 4 and higher, we show that the backward limit of type I $κ$-solutions of the Ricci flow must be a non-flat gradient shrinking Ricci soliton. This extends Perelman's previous result on backward limits of $κ$-solutions in dimension 3, in which case that the curvature operator is nonnegative (follows from Hamilton-Ivey curvature pinching estimate). The Gaussian bounds that we obtain on the fundamental solution of the conjugate heat equation under evolving metric might be of independent interest.

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Xiaodong Cao, Qi S. Zhang. 2010-06-03. The Conjugate Heat Equation and Ancient Solutions of the Ricci Flow. https://arxiv.org/abs/1006.0540

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