Filling Radius and Short Closed Geodesics of the $2$-Sphere

Sabourau, Stéphane

Bulletin de la Société Mathématique de France, Tome 132 (2004) no. 1, pp. 105-136.

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.

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.

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

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

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

[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 | Zbl

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

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

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

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

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

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

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

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

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

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

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

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

[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 | Zbl

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

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

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

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

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

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

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

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

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

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

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

[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 | Zbl

[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 | Zbl

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

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

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

[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 | Zbl

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

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

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