The length of a shortest closed geodesic on a surface of finite area

Beach, I.; Rotman, R.

doi:10.1090/proc/15194

The length of a shortest closed geodesic on a surface of finite area
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by I. Beach and R. Rotman PDF

Proc. Amer. Math. Soc. 148 (2020), 5355-5367 Request permission

Abstract:

In this paper we prove new upper bounds for the length of a shortest closed geodesic, denoted $l(M)$, on a complete, non-compact Riemannian surface $M$ of finite area $A$. We will show that $l(M) \leq 4\sqrt {2A}$ on a manifold with one end, thus improving the prior estimate of C. B. Croke, who first established that $l(M) \leq 31 \sqrt {A}$. Additionally, for a surface with at least two ends we show that $l(M) \leq 2\sqrt {2A}$, improving the prior estimate of Croke that $l(M) \leq (12+3\sqrt {2})\sqrt {A}$.

References

Frederick Justin Almgren Jr., The homotopy groups of the integral cycle groups, Topology 1 (1962), 257–299. MR 146835, DOI 10.1016/0040-9383(62)90016-2
Luca Asselle and Marco Mazzucchelli, On the existence of infinitely many closed geodesics on non-compact manifolds, Proc. Amer. Math. Soc. 145 (2017), no. 6, 2689–2697. MR 3626521, DOI 10.1090/proc/13398
Florent Balacheff, A local optimal diastolic inequality on the two-sphere, J. Topol. Anal. 2 (2010), no. 1, 109–121. MR 2646992, DOI 10.1142/S1793525310000264
Victor Bangert, Closed geodesics on complete surfaces, Math. Ann. 251 (1980), no. 1, 83–96. MR 583827, DOI 10.1007/BF01420283
Vieri Benci and Fabio Giannoni, Closed geodesics on noncompact Riemannian manifolds, C. R. Acad. Sci. Paris Sér. I Math. 312 (1991), no. 11, 857–861 (English, with French summary). MR 1108507
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
Keith Burns and Vladimir S. Matveev, Open problems and questions about geodesics, arXiv:1308.5417, 2013.
Eugenio Calabi and Jian Guo Cao, Simple closed geodesics on convex surfaces, J. Differential Geom. 36 (1992), no. 3, 517–549. MR 1189495
Jeff Cheeger and David G. Ebin, Comparison theorems in Riemannian geometry, North-Holland Mathematical Library, Vol. 9, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975. MR 0458335
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
Christopher B. Croke, Area and the length of the shortest closed geodesic, J. Differential Geom. 27 (1988), no. 1, 1–21. MR 918453
Mikhael Gromov, Filling Riemannian manifolds, J. Differential Geom. 18 (1983), no. 1, 1–147. MR 697984
James J. Hebda, Some lower bounds for the area of surfaces, Invent. Math. 65 (1981/82), no. 3, 485–490. MR 643566, DOI 10.1007/BF01396632
Antonia Jabbour and Stéphane Sabourau, Sharp upper bounds on the length of the shortest closed geodesic on complete punctured spheres of finite area, preprint, arXiv:2009.10144, 2020.
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
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, Flowers on Riemannian manifolds, Math. Z. 269 (2011), no. 1-2, 543–554. MR 2836083, DOI 10.1007/s00209-010-0749-7
Regina Rotman, Wide short geodesic loops on closed Riemannian manifolds, preprint, 2019.
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, Local extremality of the Calabi-Croke sphere for the length of the shortest closed geodesic, J. Lond. Math. Soc. (2) 82 (2010), no. 3, 549–562. MR 2739055, DOI 10.1112/jlms/jdq045
Gudlaugur Thorbergsson, Closed geodesics on non-compact Riemannian manifolds, Math. Z. 159 (1978), no. 3, 249–258. MR 493872, DOI 10.1007/BF01214574