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

Presented by: 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.

Abstract
Résumé

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.

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.

Received: 2013-02-25
Accepted: 2013-03-08
Published online: 2013-02-01

DOI: 10.1016/j.crma.2013.03.003

Author's affiliations:

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},
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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/

References
Cited by

[1] Bennett Chow On the entropy estimate for the Ricci flow on compact 2-orbifolds, J. Differ. Geom., Volume 33 (1991) no. 2, pp. 597-600

[2] Bennett Chow; Dan Knopf The Ricci Flow: An Introduction, Math. Surveys Monogr., vol. 110, Amer. Math. Soc., Providence, RI, 2004

[3] Bennett Chow; Peng Lu; Lei Ni Hamiltonʼs Ricci Flow, Grad. Stud. Math., vol. 77, Amer. Math. Soc./Science Press, Providence, RI/New York, 2006

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

[5] Richard Hamilton The Ricci flow on surfaces, Santa Cruz, CA, 1986 (Contemporary Mathematics), Volume vol. 71, American Mathematical Society, Providence, RI (1988), pp. 237-262

[6] Lei Ni The entropy formula for linear heat equation, J. Geom. Anal., Volume 14 (2004) no. 1, pp. 87-100 (Addenda: J. Geom. Anal., 14, 2, 2004, pp. 369-374)

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

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