Short geodesic loops on complete Riemannian manifolds with a finite volume

Rotman, Regina

doi:10.1090/S0002-9947-2013-05628-0

Short geodesic loops on complete Riemannian manifolds with a finite volume
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Trans. Amer. Math. Soc. 365 (2013), 2881-2894 Request permission

Abstract:

In this paper we will show that on any complete noncompact Riemannian manifold with a finite volume there exist uncountably many geodesic loops of arbitrarily small length.

References

Victor Bangert, Closed geodesics on complete surfaces, Math. Ann. 251 (1980), no. 1, 83–96. MR 583827, DOI 10.1007/BF01420283
Florent Balacheff, Sur des problèmes de la géométrie systolique, Séminaire de Théorie Spectrale et Géométrie. Vol. 22. Année 2003–2004, Sémin. Théor. Spectr. Géom., vol. 22, Univ. Grenoble I, Saint-Martin-d’Hères, 2004, pp. 71–82 (French, with French summary). MR 2136136
Yu. D. Burago and V. A. Zalgaller, Geometric inequalities, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 285, Springer-Verlag, Berlin, 1988. Translated from the Russian by A. B. Sosinskiĭ; Springer Series in Soviet Mathematics. MR 936419, DOI 10.1007/978-3-662-07441-1
Christopher B. Croke, Area and the length of the shortest closed geodesic, J. Differential Geom. 27 (1988), no. 1, 1–21. MR 918453
Christopher B. Croke and Mikhail Katz, Universal volume bounds in Riemannian manifolds, Surveys in differential geometry, Vol. VIII (Boston, MA, 2002) Surv. Differ. Geom., vol. 8, Int. Press, Somerville, MA, 2003, pp. 109–137. MR 2039987, DOI 10.4310/SDG.2003.v8.n1.a4
Matthew P. Gaffney, The conservation property of the heat equation on Riemannian manifolds, Comm. Pure Appl. Math. 12 (1959), 1–11. MR 102097, DOI 10.1002/cpa.3160120102
Mikhael Gromov, Filling Riemannian manifolds, J. Differential Geom. 18 (1983), no. 1, 1–147. MR 697984
Larry Guth, Volumes of balls in large Riemannian manifolds, Ann. of Math. (2) 173 (2011), no. 1, 51–76. MR 2753599, DOI 10.4007/annals.2011.173.1.2
Masao Maeda, The length of a closed geodesic on a compact surface, Kyushu J. Math. 48 (1994), no. 1, 9–18. MR 1269063, DOI 10.2206/kyushujm.48.9
A. Nabutovsky and R. Rotman, The length of the shortest closed geodesic on a 2-dimensional sphere, Int. Math. Res. Not. 23 (2002), 1211–1222. MR 1903953, DOI 10.1155/S1073792802110038
A. Nabutovsky and R. Rotman, Volume, diameter and the minimal mass of a stationary 1-cycle, Geom. Funct. Anal. 14 (2004), no. 4, 748–790. MR 2084979, DOI 10.1007/s00039-004-0474-7
A. Nabutovsky and R. Rotman, Curvature-free upper bounds for the smallest area of a minimal surface, Geom. Funct. Anal. 16 (2006), no. 2, 453–475. MR 2231470, DOI 10.1007/s00039-006-0559-6
Alexander Nabutovsky and Regina Rotman, The minimal length of a closed geodesic net on a Riemannian manifold with a nontrivial second homology group, Geom. Dedicata 113 (2005), 243–254. MR 2171308, DOI 10.1007/s10711-005-5177-1
Alexander Nabutovsky and Regina Rotman, The length of the second shortest geodesic, Comment. Math. Helv. 84 (2009), no. 4, 747–755. MR 2534478, DOI 10.4171/CMH/179
R. Rotman, The length of a shortest closed geodesic and the area of a 2-dimensional sphere, Proc. Amer. Math. Soc. 134 (2006), no. 10, 3041–3047. MR 2231630, DOI 10.1090/S0002-9939-06-08297-9
Regina Rotman, The length of a shortest geodesic net on a closed Riemannian manifold, Topology 46 (2007), no. 4, 343–356. MR 2321036, DOI 10.1016/j.top.2006.10.003
Regina Rotman, The length of a shortest geodesic loop at a point, J. Differential Geom. 78 (2008), no. 3, 497–519. MR 2396252
R. Rotman, Flowers on Riemannian manifolds, preprint, MPIM2006-104, available at www.mpim-bonn.mpg.de/preprints/retrieve.
Stéphane Sabourau, Filling radius and short closed geodesics of the 2-sphere, Bull. Soc. Math. France 132 (2004), no. 1, 105–136 (English, with English and French summaries). MR 2075918, DOI 10.24033/bsmf.2461
Stéphane Sabourau, Global and local volume bounds and the shortest geodesic loops, Comm. Anal. Geom. 12 (2004), no. 5, 1039–1053. MR 2103310, DOI 10.4310/CAG.2004.v12.n5.a3
Gudlaugur Thorbergsson, Closed geodesics on non-compact Riemannian manifolds, Math. Z. 159 (1978), no. 3, 249–258. MR 493872, DOI 10.1007/BF01214574
Stefan Wenger, A short proof of Gromov’s filling inequality, Proc. Amer. Math. Soc. 136 (2008), no. 8, 2937–2941. MR 2399061, DOI 10.1090/S0002-9939-08-09203-4