Paper · Jul 2, 2024
Li-Yau Conformal Volume and Colding-Minicozzi Entropy of Self-Shrinkers
As on arXiv
Abstract
We show the (normalized) Li-Yau conformal volume of a self-shrinker of mean curvature flow in Euclidean space bounds its Colding-Minicozzi entropy from below. This bound is independent of codimension and sharp on planes. As an application we verify a conjecture of Colding-Minicozzi about the entropy of closed self-shrinkers of arbitrary codimension for self-shrinkers that are topologically two-dimensional real projective planes. As part of the proof we introduce two auxiliary functionals which we call stable conformal volume and virtual entropy which should be of independent interest.
Comments: 28 pages
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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
- 6D.5Diffusions on Manifolds and Curvature
Brownian motion on a Riemannian manifold, the Bakry–Émery calculus, curvature-dimension inequalities, gradient estimates and log-Sobolev, coupling, and Brownian motion along a Ricci flow.
- 11B.2Ancient Solutions and the Harnack Inequality
Comparing curvature across space and time.
- 12A.6𝓛-Geodesics and the Variational Formulas
Perelman's §7.1 derived line by line: the first and second variation of 𝓛-length, 𝓛-Jacobi fields, adapted fields, Hamilton's Harnack expressions, and the differential inequalities for the reduced distance.