Diastolic and isoperimetric inequalities on surfaces

Balacheff, Florent; Sabourau, Stéphane

Diastolic and isoperimetric inequalities on surfaces
[Inégalités diastoliques et isopérimétriques sur les surfaces]

Annales scientifiques de l'École Normale Supérieure, Tome 43 (2010), p. 579-605 / Harvested from Numdam

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Résumé
Abstract

Nous démontrons une inégalité universelle entre la diastole, définie par un procédé de minimax sur l’espace des $1$ -cycles, et l’aire d’une surface riemannienne fermée. De manière informelle, nous prouvons que toute surface riemannienne fermée peut être balayée par une famille de multi-lacets dont les longueurs sont contrôlées par l’aire de la surface. Cette inégalité diastolique, qui repose sur une majoration de la constante de Cheeger, fournit en particulier un procédé effectif pour trouver de courtes géodésiques fermées sur une $2$ -sphère. Nous déduisons que toute surface riemannienne peut être décomposée en deux domaines de même aire dont la longueur du bord commun est majorée à l’aide de l’aire de la surface. Nous comparons également divers invariants riemanniens sur la $2$ -sphère afin de souligner le rôle spécial joué par la diastole.

We prove a universal inequality between the diastole, defined using a minimax process on the one-cycle space, and the area of closed Riemannian surfaces. Roughly speaking, we show that any closed Riemannian surface can be swept out by a family of multi-loops whose lengths are bounded in terms of the area of the surface. This diastolic inequality, which relies on an upper bound on Cheeger's constant, yields an effective process to find short closed geodesics on the two-sphere, for instance. We deduce that every Riemannian surface can be decomposed into two domains with the same area such that the length of their boundary is bounded from above in terms of the area of the surface. We also compare various Riemannian invariants on the two-sphere to underline the special role played by the diastole.

Publié le : 2010-01-01

MR 2722509 | Zbl 1226.53041

DOI : https://doi.org/10.24033/asens.2128
Classification: 53C23, 53C20, 58E10
Mots clés: constante de Cheeger, géodésiques fermées, inégalités sans courbure, diastole, inégalités isopérimétriques,

1

-cycles

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     author = {Balacheff, Florent and Sabourau, St\'ephane},
     title = {Diastolic and isoperimetric inequalities on surfaces},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     volume = {43},
     year = {2010},
     pages = {579-605},
     doi = {10.24033/asens.2128},
     mrnumber = {2722509},
     zbl = {1226.53041},
     language = {en},
     url = {http://dml.mathdoc.fr/item/ASENS_2010_4_43_4_579_0}
}

Balacheff, Florent; Sabourau, Stéphane. Diastolic and isoperimetric inequalities on surfaces. Annales scientifiques de l'École Normale Supérieure, Tome 43 (2010) pp. 579-605. doi : 10.24033/asens.2128. http://gdmltest.u-ga.fr/item/ASENS_2010_4_43_4_579_0/

Bibliographie

[1] F. J. J. Almgren, The homotopy groups of the integral cycle groups, Topology 1 (1962), 257-299. | MR 146835 | Zbl 0118.18503

[2] F. J. J. Almgren, The theory of varifolds, mimeographed notes, Princeton, 1965.

[3] F. Balacheff, Sur des problèmes de la géométrie systolique, in Séminaire de Théorie Spectrale et Géométrie. Vol. 22. Année 2003-2004, Sémin. Théor. Spectr. Géom. 22, Univ. Grenoble I, 2004, 71-82. | Numdam | MR 2136136 | Zbl 1083.53043

[4] F. Balacheff, A local optimal diastolic inequality on the two-sphere, preprint arXiv:0811.0330. | MR 2646992 | Zbl 1193.53105

[5] R. Brooks, The spectral geometry of a tower of coverings, J. Differential Geom. 23 (1986), 97-107. | MR 840402 | Zbl 0576.58033

[6] E. Calabi & J. G. Cao, Simple closed geodesics on convex surfaces, J. Differential Geom. 36 (1992), 517-549. | MR 1189495 | Zbl 0768.53019

[7] J. Cheeger, A lower bound for the smallest eigenvalue of the Laplacian, in Problems in analysis (Papers dedicated to Salomon Bochner, 1969), Princeton Univ. Press, 1970, 195-199. | MR 402831 | Zbl 0212.44903

[8] Y. Colin De Verdière & A. Marin, Triangulations presque équilatérales des surfaces, J. Differential Geom. 32 (1990), 199-207. | MR 1064872 | Zbl 0705.53021

[9] C. B. Croke, Area and the length of the shortest closed geodesic, J. Differential Geom. 27 (1988), 1-21. | MR 918453 | Zbl 0642.53045

[10] H. Federer, Geometric measure theory, Grundl. Math. Wiss. 153, Springer New York Inc., New York, 1969. | MR 257325 | Zbl 0176.00801

[11] W. H. Fleming, Flat chains over a finite coefficient group, Trans. Amer. Math. Soc. 121 (1966), 160-186. | MR 185084 | Zbl 0136.03602

[12] M. Gromov, Filling Riemannian manifolds, J. Differential Geom. 18 (1983), 1-147. | MR 697984 | Zbl 0515.53037

[13] M. Gromov, Width and related invariants of Riemannian manifolds, Astérisque 163-164 (1988), 93-109. | MR 999972 | Zbl 0684.53036

[14] L. Guth, Lipschitz maps from surfaces, Geom. Funct. Anal. 15 (2005), 1052-1099. | MR 2221158 | Zbl 1101.53021

[15] L. Guth, The width-volume inequality, Geom. Funct. Anal. 17 (2007), 1139-1179. | MR 2373013 | Zbl 1141.53039

[16] M. G. Katz, The filling radius of two-point homogeneous spaces, J. Differential Geom. 18 (1983), 505-511. | MR 723814 | Zbl 0529.53032

[17] M. G. Katz & S. Sabourau, Entropy of systolically extremal surfaces and asymptotic bounds, Ergodic Theory Dynam. Systems 25 (2005), 1209-1220. | MR 2158402 | Zbl 1097.53027

[18] W. Klingenberg, Lectures on closed geodesics, Grundl. der Math. Wiss. 230, Springer, 1978. | MR 478069 | Zbl 0397.58018

[19] P. Li & S. T. Yau, A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces, Invent. Math. 69 (1982), 269-291. | MR 674407 | Zbl 0503.53042

[20] A. Nabutovsky & R. Rotman, The length of the shortest closed geodesic on a 2-dimensional sphere, Int. Math. Res. Not. 2002 (2002), 1211-1222. | MR 1903953 | Zbl 1003.53030

[21] P. Papasoglu, Cheeger constants of surfaces and isoperimetric inequalities, Trans. Amer. Math. Soc. 361 (2009), 5139-5162. | MR 2515806 | Zbl 1183.53030

[22] J. T. Pitts, Regularity and singularity of one dimensional stationary integral varifolds on manifolds arising from variational methods in the large, in Symposia Mathematica, Vol. XIV (Convegno di Teoria Geometrica dell'Integrazione e Varietà Minimali, INDAM, Roma, Maggio 1973), Academic Press, 1974, 465-472. | MR 423172 | Zbl 0312.49017

[23] J. T. Pitts, Existence and regularity of minimal surfaces on Riemannian manifolds, Mathematical Notes 27, Princeton Univ. Press, 1981. | MR 626027 | Zbl 0462.58003

[24] G. De Rham, Variétés différentiables. Formes, courants, formes harmoniques, Actualités Sci. Ind., no. 1222 = Publ. Inst. Math. Univ. Nancago III, Hermann et Cie, Paris, 1955. | Zbl 0065.32401

[25] R. Rotman, The length of a shortest closed geodesic on a two-dimensional sphere and coverings by metric balls, Geom. Dedicata 110 (2005), 143-157. | MR 2136024 | Zbl 1080.53031

[26] R. Rotman, The length of a shortest closed geodesic and the area of a 2-dimensional sphere, Proc. Amer. Math. Soc. 134 (2006), 3041-3047. | MR 2231630 | Zbl 1098.53035

[27] S. Sabourau, Sur quelques problèmes de la géométrie systolique, thèse de doctorat, Univ. de Montpellier II, 2001.

[28] S. Sabourau, Filling radius and short closed geodesics of the 2-sphere, Bull. Soc. Math. France 132 (2004), 105-136. | Numdam | MR 2075918 | Zbl 1064.53020

[29] S. Sabourau, Local extremality of the Calabi-Croke sphere for the length of the shortest closed geodesic, preprint arXiv:0907.2223. | MR 2739055 | Zbl 1208.53052

[30] A. Treibergs, Estimates of volume by the length of shortest closed geodesics on a convex hypersurface, Invent. Math. 80 (1985), 481-488. | MR 791672 | Zbl 0563.53035

[31] P. C. Yang & S. T. Yau, Eigenvalues of the Laplacian of compact Riemann surfaces and minimal submanifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 7 (1980), 55-63. | Numdam | MR 577325 | Zbl 0446.58017