Paper · Jan 2, 2020
On the Shannon entropy power on Riemannian manifolds and Ricci flow
As on arXiv
Abstract
In this paper, we prove the concavity of the Shannon entropy power for the heat equation associated with the Laplacian or the Witten Laplacian on complete Riemannian manifolds with suitable curvature-dimension condition and on compact super Ricci flows. Under suitable curvature-dimension condition, we prove that the rigidity models of the Shannon entropy power are Einstein or quasi Einstein manifolds with Hessian solitons. Moreover, we prove the convexity of the Shannon entropy power for the conjugate heat equation introduced by G. Perelman on Ricci flow and that the corresponding rigidity models are the shrinking Ricci solitons. As an application, we prove the entropy isoperimetric inequality on complete Riemannian manifolds with non-negative (Bakry-Emery) Ricci curvature and the maximal volume growth condition.
- Submitted
- Jan 2, 2020
- Subjects
- math.DG
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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
- 6A.3The Heat Equation on ℝⁿ
The heat kernel, smoothing, uniqueness, Brownian motion, and a Gaussian that runs to Perelman.
- 6A.4Fourier Methods and Scale Space
Fourier's own method on the circle and the line, smoothing as the decay of high frequencies, and the heat equation as the mathematics of blurring images.
- 6A.12Entropy, Information and Diffusion
Entropy along the heat flow, the log-Sobolev inequality and Li–Yau: the ancestors of Perelman’s entropy.