An entropy formula relating Hamiltonʼs surface entropy and Perelmanʼs $ \mathcal{W}$ entropy

Hongxin Guo

doi:10.1016/j.crma.2013.03.003

Differential Geometry

An entropy formula relating Hamiltonʼs surface entropy and Perelmanʼs

W

entropy
[Une formule dʼentropie reliant lʼentropie de Hamilton des surfaces et lʼentropie

W

de Perelman]

Présenté par : the Editorial Board

Hongxin Guo ¹

¹ School of Mathematics and Information Science, Wenzhou University, Wenzhou, Zhejiang 325035, China

Comptes Rendus. Mathématique, Volume 351 (2013) no. 3-4, pp. 115-118.

Résumé
Abstract

Dans cette note, à partir de la formule de Hamilton pour lʼentropie des surfaces, nous construisons une formule dʼentropie de type Perelman pour le flot de Ricci sur une surface fermée à courbure positive. De même que pour lʼentropie $W$ de Perelman, le point critique de notre entropie est le soliton gradient décroissant, bien quʼil nʼy ait pas ici dʼéquation de la chaleur qui soit mise en jeu. Ceci démontre une relation étroite entre lʼentropie de Hamilton et lʼentropie $W$ de Perelman sur les surfaces fermées.

In this note, based on Hamiltonʼs surface entropy formula, we construct an entropy formula of Perelmanʼs type for the Ricci flow on a closed surface with positive curvature. Similar to Perelmanʼs $W$ entropy, the critical point of our entropy is the gradient shrinking soliton; however, there is no conjugate heat equation involved. This shows a close relation between Hamiltonʼs entropy and Perelmanʼs $W$ entropy on closed surfaces.

Reçu le : 2013-02-25
Accepté le : 2013-03-08
Publié le : 2013-02-01

DOI : 10.1016/j.crma.2013.03.003

Affiliations des auteurs :

Hongxin Guo ¹

¹ School of Mathematics and Information Science, Wenzhou University, Wenzhou, Zhejiang 325035, China

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     author = {Hongxin Guo},
     title = {An entropy formula relating {Hamilton's} surface entropy and {Perelman's} $ \mathcal{W}$ entropy},
     journal = {Comptes Rendus. Math\'ematique},
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Hongxin Guo. An entropy formula relating Hamiltonʼs surface entropy and Perelmanʼs $ \mathcal{W}$ entropy. Comptes Rendus. Mathématique, Volume 351 (2013) no. 3-4, pp. 115-118. doi : 10.1016/j.crma.2013.03.003. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2013.03.003/

Bibliographie
Cité par

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[4] Hongxin Guo, Robert Philipowski, Anton Thalmaier, Entropy and lowest eigenvalue on evolving manifolds, Pacific J. Math., in press.

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[7] Grisha Perelman The entropy formula for the Ricci flow and its geometric applications | arXiv

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