Pseudolocality of $G_2$-Laplacian Flow

Weiyong He; Longhuan Yan

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arXiv · 2610.38408

Pseudolocality of $G_2$-Laplacian Flow

Weiyong He·Longhuan Yan

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Abstract

In this paper, we give a new proof of Perelman's pseudolocality theorem for Ricci flow, which does not rely on Perelman's Harnack-type inequality. The key step is an estimate relating initial local $μ$-entropy to pointed Nash entropy at later spacetime points. The argument also works for other reasonable geometric flows, like the $G_2$-Laplacian flow, at a scale depending on its deviation from Ricci flow and its scalar curvature lower bound. As a consequence, we can also prove the quantitative rigidity results concerning pointed Nash entropy due to Hein-Naber and Bamler for such geometric flows. In the appendix, we also show that the other results in Bamler's work (arXiv:2008.07093), when suitably modified, remain valid for Laplacian flow with bounded torsion. These include log-Sobolev/Poincaré inequality for conjugate heat kernel measure, heat kernel estimates, and dependence of the pointed Nash entropy on its basepoint.

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Weiyong He, Longhuan Yan. 2026-10-28. Pseudolocality of $G_2$-Laplacian Flow. https://arxiv.org/abs/2610.38408

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