Filling Radius and Short Closed Geodesics of the 2-Sphere
Bulletin de la Société Mathématique de France, Volume 132 (2004) no. 1, p. 105-136

We show that the length of the shortest nontrivial curve among the simple closed geodesics of index zero or one and the figure-eight geodesics of null index provides a lower bound on the area and the diameter of the Riemannian 2-spheres.

Nous montrons que la longueur de la plus courte courbe non triviale parmi les géodésiques simples fermées d'indice zéro ou un et les géodésiques en huit d'indice nul fournit une minoration sur l'aire et le diamètre des deux-sphères riemanniennes.

DOI : https://doi.org/10.24033/bsmf.2461
Classification:  53C20
Keywords: filling radius, closed geodesics, 1-cycles
@article{BSMF_2004__132_1_105_0,
     author = {Sabourau, St\'ephane},
     title = {Filling Radius and Short Closed Geodesics of the $2$-Sphere},
     journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {132},
     number = {1},
     year = {2004},
     pages = {105-136},
     doi = {10.24033/bsmf.2461},
     zbl = {1064.53020},
     mrnumber = {2075918},
     language = {en},
     url = {http://www.numdam.org/item/BSMF_2004__132_1_105_0}
}
Sabourau, Stéphane. Filling Radius and Short Closed Geodesics of the $2$-Sphere. Bulletin de la Société Mathématique de France, Volume 132 (2004) no. 1, pp. 105-136. doi : 10.24033/bsmf.2461. http://www.numdam.org/item/BSMF_2004__132_1_105_0/

[1] R. Accola - « Differential and extremal lengths on Riemannian surfaces », Proc. Math. Acad. Sci. USA 46 (1960), p. 83-96. | MR 118829

[2] F. Almgren - « The homotopy groups of the integral cycle groups », Topology 1 (1960), p. 257-299. | MR 146835 | Zbl 0118.18503

[3] I. Babenko - « Asymptotic invariants of smooth manifolds », Russian Acad. Sci. Izv. Math. 41 (1993), p. 1-38. | MR 1208148 | Zbl 0812.57022

[4] V. Bangert & M. Katz - « Riemannian manifolds with harmonic 1-forms of constant norms », Preprint.

[5] -, « Stable systolic inequalities and cohomology products », Comm. Pure Appl. Math. 56 (2003). | MR 1990484 | Zbl 1038.53031

[6] C. Bavard - « Inégalité isosystolique pour la bouteille de Klein », Math. Ann. 274 (1986), p. 439-441. | MR 842624 | Zbl 0578.53032

[7] M. Berger - « Systole et applications selon Gromov », Sémin. Bourbaki, Astérisque, vol. 216, Soc. Math. France, 1993, p. 279-310. | Numdam | MR 1246401 | Zbl 0789.53040

[8] C. Blatter - « Über extremallägen auf geschlossen flächen », Comment. Math. Helv. 35 (1961), p. 55-62. | MR 131539 | Zbl 0107.28404

[9] Y. Burago & V. Zalgaller - Geometric inequalities, Springer, 1988. | MR 936419 | Zbl 0633.53002

[10] E. Calabi & J. Cao - « Simple closed geodesics on convex surfaces », J. Diff. Geom. 36 (1992), p. 517-549. | MR 1189495 | Zbl 0768.53019

[11] J. Cheeger & D. Ebin - Comparaison theorems in Riemannian geometry, North-Holland, Amsterdam, 1975. | MR 458335 | Zbl 0309.53035

[12] C. Croke - « Area and the length of shortest closed geodesic », J. Diff. Geom. 27 (1988), p. 1-22. | MR 918453 | Zbl 0642.53045

[13] C. Croke & M. Katz - « Universal volume bounds in Riemannian manifolds », Surveys in Differential Geometry, vol. 8, to appear. | MR 2039987 | Zbl 1051.53026

[14] H. Federer - Geometric measure theory, Springer-Verlag, New York, 1969. | MR 257325 | Zbl 0874.49001

[15] S. Frankel & M. Katz - « Morse landscape of a Riemannian disk », Ann. Inst. Fourier 43 (1993), p. 503-507. | Numdam | MR 1220281 | Zbl 0780.53035

[16] M. Grayson - « Shortening embedded curves », Ann. Math. 129 (1989), p. 71-111. | MR 979601 | Zbl 0686.53036

[17] M. Gromov - « Filling Riemannian manifolds », J. Diff. Geom. 18 (1983), p. 1-147. | MR 697984 | Zbl 0515.53037

[18] -, « Systoles and intersystolic inequalities », Actes de la Table Ronde de Géométrie Différentielle (Luminy, 1992), Sémin. Congr., vol. 1, Soc. Math. France, 1996, p. 291-363. | MR 1427763 | Zbl 0877.53002

[19] -, Metric structures for Riemannian and non-Riemannian spaces, Progr. in Mathematics, vol. 152, Birkhäuser, Boston, 1999. | MR 1699320

[20] J. Hass & P. Scott - « Shortening curves on surfaces », Topology 33 (1994), p. 25-43. | MR 1259513 | Zbl 0798.58019

[21] J. Hebda - « Some lower bounds for the area of surfaces », Invent. Math. 65 (1982), p. 485-491. | MR 643566 | Zbl 0482.53028

[22] -, « The collars of a Riemannian manifold and stable isosystolic inequalities », Pacific J. Math. 121 (1986), p. 339-356. | MR 819193 | Zbl 0607.53043

[23] M. Katz - « The filling radius of two-point homogeneous spaces », J. Diff. Geom. 18 (1983), p. 505-511. | MR 723814 | Zbl 0529.53032

[24] M. Katz, M. Kreck & A. Suciu - « Free abelian covers, short loops, stable length and systolic inequalities », Preprint. | Zbl 1134.53019

[25] W. Klingenberg - Lectures on closed geodesics, Appendix, Grundlehren Math. Wiss., vol. 230, Springer-Verlag, Berlin, 1978. | MR 478069 | Zbl 0397.58018

[26] M. Maeda - « The length of a closed geodesic on a compact surface », Kyushu J. Math. 48 (1994), no. 1, p. 9-18. | MR 1269063 | Zbl 0818.53064

[27] F. Morgan - Geometric measure theory. A beginner's guide, 2nd éd., Academic Press, 1995. | MR 1326605 | Zbl 0671.49043

[28] A. Nabutovsky & R. Rotman - « Volume, diameter and the minimal mass of a stationnary 1-cycle », Preprint. | MR 2084979 | Zbl 1073.53057

[29] -, « The length of the shortest closed geodesic on a 2-dimensional sphere », Int. Math. Res. Not. 23 (2002), p. 1211-1222. | MR 1903953 | Zbl 1003.53030

[30] -, « Upper bounds on the length of a shortest closed geodesic and quantitative Hurewicz theorem », J. Eur. Math. Soc. 5 (2003), p. 203-244. | MR 2002213 | Zbl 1039.53042

[31] J. Pitts - Regularity and singularity of one dimensional stational integral varifolds on manifolds arising from variational methods in the large, Symposia Mathematics, vol. XIV, Roma, Italy, 1974. | MR 423172 | Zbl 0312.49017

[32] -, Existence and regularity of minimal surfaces on Riemannian manifolds, Math. Notes, vol. 27, Princeton University Press, 1981. | MR 626027 | Zbl 0462.58003

[33] P. Pu - « Some inequalities in certain nonorientable Riemannian manifolds », Pacific J. Math. 2 (1952), p. 55-71. | MR 48886 | Zbl 0046.39902

[34] G. De Rham - Differentiable manifolds, Grundlehren Math. Wiss., vol. 266, Springer-Verlag, Berlin, 1984. | MR 760450 | Zbl 0534.58003

[35] R. Rotman - « Upper bounds on the length of the shortest closed geodesic on simply connected manifolds », Math. Z. 233 (2000), p. 365-398. | MR 1743442 | Zbl 0960.53026

[36] T. Sakai - « A proof of the isosystolic inequality for the Klein bottle », Proc. Amer. Math. Soc. 104 (1988), p. 589-590. | MR 962833 | Zbl 0692.53019

[37] B. White - « A strong minimax property of nondegenerate minimal submanifolds », J. reine angew. Math. 457 (1994), p. 203-218. | MR 1305283 | Zbl 0808.49037

[38] F. Wilhelm - « On radius, systole and positive Ricci curvature », Math. Z. 218 (1995), p. 597-602. | MR 1326989 | Zbl 0823.53033