Paper · Aug 17, 2026
Regularizing estimates for positive solutions of the heat equation under geometric flows
As on arXiv
Abstract
We study higher-order global estimates for the heat equation on Riemannian manifolds, both for static metrics and for metrics evolving under the Ricci flow. Under minimal geometric assumptions, we derive first-order regularizing estimates for log-solutions of the heat equation, together with upper second-order bounds with explicit constants. Our quantitative approach is based on integral duality methods proposed by L.\ C.\ Evans, J.-M.\ Lasry and P.-L.\ Lions in different settings.
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Background reading
The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.
In the guidebook
- 5A.5Uniformization and the Two-Dimensional Ricci Flow
Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.
- 10A.12Curve Shortening and Minimal Discs
How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.
- 11A.1The Equation and Its First Solutions
A heat equation for the metric: spheres, cylinders and scaling.