Smoothening cone points with Ricci flow

Ramos, Daniel

doi:10.24033/bsmf.2700

Smoothening cone points with Ricci flow
[Lissage de points coniques avec flot de Ricci]

Ramos, Daniel

Bulletin de la Société Mathématique de France, Tome 143 (2015) no. 4, pp. 619-633

Abstract (VO)
Résumé

We consider Ricci flow on a closed surface with cone points. The main result is: given a (nonsmooth) cone metric $g_{0}$ over a closed surface there is a smooth Ricci flow $g (t)$ defined for $(0, T]$ , with curvature unbounded above, such that $g (t)$ tends to $g_{0}$ as $t \to 0$ . This result means that Ricci flow provides a way for instantaneously smoothening cone points. We follow the argument of P. Topping in [Int. Math. Res. Not. 2012 (2012)] modifying his reasoning for cusps of negative curvature; in that sense we can consider cusps as a limiting zero-angle cone, and we generalize to any angle between 0 and $2 π$ .

On considère un flot de Ricci sur une surface fermée avec des points coniques. Le résultat principal est : étant donné une métrique conique $g_{0}$ (non lisse) sur une surface fermée, il existe un flot de Ricci lisse $g (t)$ défini pour $(0, T]$ , avec courbure non bornée supérieurement, tel que $g (t)$ tend vers $g_{0}$ quand $t \to 0$ . Cet résultat implique que le flot de Ricci donne une méthode pour lisser instantanément des points coniques. On suit un argument de P. Topping dans [Int. Math. Res. Not. 2012 (2012)] en modifiant son raisonnement pour les cusps de courbure négative; en ce sens, on peut considérer les cusps comme un cas limite de points coniques d'angle zéro, et nous généralisons à un angle quelconque entre 0 et $2 π$ .

Publié le : 2015-07-21

MR Zbl

DOI : 10.24033/bsmf.2700

Classification : 53C44
Keywords: Ricci flow, conic singularities.
Mots-clés : Flot de Ricci, singularités coniques.

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     title = {Smoothening cone points with {Ricci} flow},
     journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
     pages = {619--633},
     year = {2015},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {143},
     number = {4},
     doi = {10.24033/bsmf.2700},
     mrnumber = {3450497},
     zbl = {1335.53089},
     language = {en},
     url = {https://www.numdam.org/articles/10.24033/bsmf.2700/}
}

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Ramos, Daniel. Smoothening cone points with Ricci flow. Bulletin de la Société Mathématique de France, Tome 143 (2015) no. 4, pp. 619-633. doi: 10.24033/bsmf.2700

Bibliographie
Cité par

Chen, Xiuxiong; Lu, Peng; Tian, Gang A note on uniformization of Riemann surfaces by Ricci flow, Proc. Amer. Math. Soc., Volume 134 (2006), pp. 3391-3393 (ISSN: 0002-9939) | MR | Zbl | DOI

Chow, Bennett On the entropy estimate for the Ricci flow on compact 2-orbifolds, J. Differential Geom., Volume 33 (1991), pp. 597-600 http://projecteuclid.org/euclid.jdg/1214446332 (ISSN: 0022-040X) | MR | Zbl

Chow, Bennett The Ricci flow on the 2-sphere, J. Differential Geom., Volume 33 (1991), pp. 325-334 http://projecteuclid.org/euclid.jdg/1214446319 (ISSN: 0022-040X) | MR | Zbl

Chow, Bennett; Knopf, Dan The Ricci flow: an introduction, Mathematical Surveys and Monographs, 110, Amer. Math. Soc., Providence, RI, 2004, 325 pages (ISBN: 0-8218-3515-7) | MR | Zbl | DOI

Chow, Bennett; Wu, Lang-Fang The Ricci flow on compact 2-orbifolds with curvature negative somewhere, Comm. Pure Appl. Math., Volume 44 (1991), pp. 275-286 (ISSN: 0010-3640) | MR | Zbl | DOI

Giesen, Gregor; Topping, Peter M. Existence of Ricci flows of incomplete surfaces, Comm. Partial Differential Equations, Volume 36 (2011), pp. 1860-1880 (ISSN: 0360-5302) | MR | Zbl | DOI

Hamilton, Richard S., Mathematics and general relativity (Santa Cruz, CA, 1986) (Contemp. Math.), Volume 71, Amer. Math. Soc., Providence, RI, 1988, pp. 237-262 | MR | Zbl | DOI

Mazzeo, Rafe; Rubinstein, Yanir A.; Sesum, Natasa Ricci flow on surfaces with conic singularities, Anal. PDE, Volume 8 (2015), pp. 839-882 (ISSN: 2157-5045) | MR | Zbl | DOI

Richard, Thomas Canonical smoothing of compact Alexandrov surfaces via Ricci flow (preprint arXiv:1204.5461 ) | MR | Zbl | Numdam

Simon, Miles Ricci flow of non-collapsed three manifolds whose Ricci curvature is bounded from below, J. reine angew. Math., Volume 662 (2012), pp. 59-94 (ISSN: 0075-4102) | MR | Zbl | DOI

Topping, Peter M. Uniqueness and nonuniqueness for Ricci flow on surfaces: reverse cusp singularities, Int. Math. Res. Not., Volume 2012 (2012), pp. 2356-2376 (ISSN: 1073-7928) | MR | Zbl | DOI

Wu, Lang-Fang The Ricci flow on 2-orbifolds with positive curvature, J. Differential Geom., Volume 33 (1991), pp. 575-596 http://projecteuclid.org/euclid.jdg/1214446331 (ISSN: 0022-040X) | MR | Zbl

Yin, Hao Ricci flow on surfaces with conical singularities, J. Geom. Anal., Volume 20 (2010), pp. 970-995 (ISSN: 1050-6926) | MR | Zbl | DOI

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