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Closed geodesics on reversible Finsler 2-spheres. (English) Zbl 1496.53058

Let \((M,F)\) be a closed oriented surface endowed with a reversible Finsler metric \(F\), \(\mathbb S^1\) be the unit circle, and \(\mathrm{Emb}(\mathbb S^1,M)\) be the set of smooth \(M\)-valued embedded loops \(\gamma\) on \(\mathbb S^1\). Given \(J\in \mathrm{End}(TM)\), the positive normal to \(\gamma\in \mathrm{Emb}(\mathbb S^1,M)\) is defined by \(N_{\gamma_t}(u)=\displaystyle\frac{J_{\dot{\gamma_t}}(u)}{\|\dot{\gamma_t}(u)\|}\cdot\) The authors consider a one-parameter family of curves \(\gamma_t\in \mathrm{Emb}(\mathbb S^1,M)\) solutions of \[\partial_t\gamma_t(u)=\omega_t(u)N_{\gamma_t}(u),\tag{1}\] where \(\omega_t(u)\) is an explicit expression given in terms of the partial derivatives of \(F\) with respect to some local coordinates on \(M\), \(N_{\gamma_t}(u)\), and on the norm of \(\dot{\gamma_t}\). Then, they state that if \(\gamma_t\) is the solution of (1) with initial condition \(\gamma_0\), then there is a unique \(\mathrm{Emb}(\mathbb S^1,M)\)-valued continuous map \(\varphi\) on an open neighborhood of \(\{0\}\times\mathrm{Emb}(\mathbb S^1,M)\) such that \(\varphi(t,\gamma_0)=\varphi_t(\gamma_0)=\gamma_t\). Furthermore, \(\varphi_t(\gamma\circ\theta)=\varphi_t(\gamma)\circ\theta\) for \(\theta\in Diff(\mathbb S^1)\), \(\frac{d}{dt}L(\varphi_t(\gamma))\le 0\) for \(\gamma\in \mathrm{Emb}(\mathbb S^1,M)\) where \(L\) is the Finsler length functional on \(\mathrm{Emb}(\mathbb S^1,M)\) defined as \(L(\gamma)=\displaystyle\int_0^1F(\gamma(t),\dot{\gamma_t}(u))du\), and if \(l_\gamma=\displaystyle\lim_{t\to\tau_\gamma}L(\varphi_t(\gamma))>0\), then \(\tau_\gamma=\infty\) (Theorem 2.1). Also, they state that every reversible Finsler two-sphere \((\mathbb S^2,F)\) has at least three explicit geometrically distinct simple closed geodesics (Theorem 1.3).

MSC:

53C22 Geodesics in global differential geometry
58E10 Variational problems in applications to the theory of geodesics (problems in one independent variable)
53E99 Geometric evolution equations
53C60 Global differential geometry of Finsler spaces and generalizations (areal metrics)
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[1] Asselle, L.; Mazzucchelli, M., Closed geodesics with local homology in maximal degree on non-compact manifolds, Differ. Geom. Appl., 58, 17-51 (2018) · Zbl 1387.53054 · doi:10.1016/j.difgeo.2017.11.007
[2] Abbondandolo, A.; Macarini, L.; Paternain, GP, On the existence of three closed magnetic geodesics for subcritical energies, Comment. Math. Helv., 90, 1, 155-193 (2015) · Zbl 1316.53042 · doi:10.4171/CMH/350
[3] Angenent, S., Parabolic equations for curves on surfaces. I. Curves with \(p\)-integrable curvature, Ann. Math. (2), 132, 3, 451-483 (1990) · Zbl 0789.58070 · doi:10.2307/1971426
[4] Angenent, S., Parabolic equations for curves on surfaces. II. Intersections, blow-up and generalized solutions, Ann. Math. (2), 133, 1, 171-215 (1991) · Zbl 0749.58054 · doi:10.2307/2944327
[5] Angenent, S., Self-intersecting geodesics and entropy of the geodesic flow, Acta Math. Sin. (Engl. Ser.), 24, 12, 1949-1952 (2008) · Zbl 1162.53030 · doi:10.1007/s10114-008-6439-2
[6] Ballmann, W.: Der Satz von Lusternik und Schnirelmann, Beiträge zur Differentialgeometrie, Heft 1, Bonner Math. Schriften, vol. 102, pp. 1-25. University of Bonn, Bonn (1978) · Zbl 0394.53027
[7] Bangert, V., Closed geodesics on complete surfaces, Math. Ann., 251, 1, 83-96 (1980) · Zbl 0422.53024 · doi:10.1007/BF01420283
[8] Bangert, V., On the existence of closed geodesics on two-spheres, Int. J. Math., 4, 1, 1-10 (1993) · Zbl 0791.53048 · doi:10.1142/S0129167X93000029
[9] Birkhoff, GD, Dynamical Systems (1966), Providence: American Mathematical Society, Providence · Zbl 0171.05402
[10] Bangert, V.; Long, Y., The existence of two closed geodesics on every Finsler 2-sphere, Math. Ann., 346, 2, 335-366 (2010) · Zbl 1187.53040 · doi:10.1007/s00208-009-0401-1
[11] Bott, R., On the iteration of closed geodesics and the Sturm intersection theory, Commun. Pure Appl. Math., 9, 171-206 (1956) · Zbl 0074.17202 · doi:10.1002/cpa.3160090204
[12] Ballmann, W.; Thorbergsson, G.; Ziller, W., Closed geodesics on positively curved manifolds, Ann. Math., 116, 2, 213-247 (1982) · Zbl 0495.58010 · doi:10.2307/2007062
[13] Chang, K-C, Infinite-Dimensional Morse Theory and Multiple Solution Problems (1993), Boston: Birkhäuser Boston Inc., Boston · Zbl 0779.58005 · doi:10.1007/978-1-4612-0385-8
[14] Caponio, E.; Javaloyes, MÁ; Masiello, A., On the energy functional on Finsler manifolds and applications to stationary spacetimes, Math. Ann., 351, 2, 365-392 (2011) · Zbl 1228.53052 · doi:10.1007/s00208-010-0602-7
[15] Frauenfelder, U.; Lange, C.; Suhr, S., A Hamiltonian version of a result of Gromoll and Grove, Ann. Inst. Fourier (Grenoble), 69, 1, 409-419 (2019) · Zbl 1423.53101 · doi:10.5802/aif.3247
[16] Franks, J., Geodesics on \(S^2\) and periodic points of annulus homeomorphisms, Invent. Math., 108, 2, 403-418 (1992) · Zbl 0766.53037 · doi:10.1007/BF02100612
[17] Gromoll, D., Grove, K.: On metrics on \(S^2\) all of whose geodesics are closed. Invent. Math. 65(1), 175-177 (1981/1982) · Zbl 0477.53044
[18] Gromoll, D.; Meyer, W., On differentiable functions with isolated critical points, Topology, 8, 361-369 (1969) · Zbl 0212.28903 · doi:10.1016/0040-9383(69)90022-6
[19] Gromoll, D.; Meyer, W., Periodic geodesics on compact Riemannian manifolds, J. Differ. Geom., 3, 493-510 (1969) · Zbl 0203.54401 · doi:10.4310/jdg/1214429070
[20] Grayson, M. A., Shortening embedded curves, Ann. Math. (2), 129, 1, 71-111 (1989) · Zbl 0686.53036 · doi:10.2307/1971486
[21] Hadamard, M., Les surfaces à courbures opposées et leurs lignes géodésiques, J. Math. Pures Appl., 5, 4, 27-74 (1898) · JFM 29.0522.01
[22] Hingston, N., On the growth of the number of closed geodesics on the two-sphere, Int. Math. Res. Not., 1993, 9, 253-262 (1993) · Zbl 0809.53053 · doi:10.1155/S1073792893000285
[23] Hass, J.; Scott, P., Shortening curves on surfaces, Topology, 33, 1, 25-43 (1994) · Zbl 0798.58019 · doi:10.1016/0040-9383(94)90033-7
[24] Irie, K.; Marques, FC; Neves, A., Density of minimal hypersurfaces for generic metrics, Ann. Math. (2), 187, 3, 963-972 (2018) · Zbl 1387.53083 · doi:10.4007/annals.2018.187.3.8
[25] Jost, J., A nonparametric proof of the theorem of Lusternik and Schnirelman, Arch. Math. (Basel), 53, 5, 497-509 (1989) · Zbl 0676.58018 · doi:10.1007/BF01324725
[26] Jost, J., Correction to: “A nonparametric proof of the theorem of Lusternik and Schnirelman”, Arch. Math. (Basel), 56, 6, 624 (1991) · Zbl 0724.58020 · doi:10.1007/BF01246779
[27] Katok, AB, Ergodic perturbations of degenerate integrable Hamiltonian systems, Izv. Akad. Nauk SSSR Ser. Mat., 37, 539-576 (1973) · Zbl 0316.58010
[28] Klingenberg, W.P.A.: Riemannian Geometry, 2nd edn. De Gruyter Studies in Mathematics, vol. 1. Walter de Gruyter & Co., Berlin (1995) · Zbl 0911.53022
[29] Ljusternik, L.A.: The Topology of the Calculus of Variations in the Large. Translated from the Russian by J. M. Danskin. Translations of Mathematical Monographs, vol. 16. American Mathematical Society, Providence (1966) · Zbl 0154.37003
[30] Lusternik, L.; Schnirelmann, L., Existence de trois géodésiques fermées sur toute surface de genre 0, C. R. Acad. Sci. Paris, 188, 534-536 (1929) · JFM 55.0316.01
[31] Lu, G., Methods of infinite dimensional Morse theory for geodesics on Finsler manifolds, Nonlinear Anal., 113, 230-282 (2015) · Zbl 1327.58016 · doi:10.1016/j.na.2014.09.016
[32] Marques, FC, Abundance of minimal surfaces, Jpn. J. Math., 14, 2, 207-229 (2019) · Zbl 1423.53073 · doi:10.1007/s11537-019-1839-x
[33] Mazzucchelli, M., Symplectically degenerate maxima via generating functions, Math. Z., 275, 3-4, 715-739 (2013) · Zbl 1288.37015 · doi:10.1007/s00209-013-1157-6
[34] Mazzucchelli, M., The Morse index of Chaperon’s generating families, Publ. Mat. Urug., 16, 81-125 (2016) · Zbl 1372.37034
[35] Mantegazza, C.; Martinazzi, L., A note on quasilinear parabolic equations on manifolds, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 11, 4, 857-874 (2012) · Zbl 1272.35123
[36] Morse, M.: The Calculus of Variations in the Large. American Mathematical Society Colloquium Publications, vol. 18. American Mathematical Society, Providence (1996). Reprint of the 1932 original · JFM 58.0537.01
[37] Mazzucchelli, M.; Suhr, S., A characterization of Zoll Riemannian metrics on the 2-sphere, Bull. Lond. Math. Soc., 50, 997-1006 (2018) · Zbl 1405.53063 · doi:10.1112/blms.12200
[38] Neumann, WD, Generalizations of the Poincaré Birkhoff fixed point theorem, Bull. Aust. Math. Soc., 17, 3, 375-389 (1977) · Zbl 0372.54041 · doi:10.1017/S0004972700010650
[39] Oaks, JA, Singularities and self-intersections of curves evolving on surfaces, Indiana Univ. Math. J., 43, 3, 959-981 (1994) · Zbl 0835.53048 · doi:10.1512/iumj.1994.43.43042
[40] Poincaré, H., Sur les lignes géodésiques des surfaces convexes, Trans. Am. Math. Soc., 6, 237-274 (1905) · JFM 36.0669.01
[41] Rademacher, H.-B.: Morse-Theorie und geschlossene Geodätische, Bonner Math. Schriften, vol. 229. University of Bonn, Bonn (1992) · Zbl 0826.58012
[42] Taimanov, IA, On the existence of three nonintersecting closed geodesics on manifolds that are homeomorphic to the two-dimensional sphere, Izv. Ross. Akad. Nauk Ser. Mat., 56, 3, 605-635 (1992) · Zbl 0771.53027
[43] Vigué-Poirrier, M.; Sullivan, D., The homology theory of the closed geodesic problem, J. Differ. Geom., 11, 4, 633-644 (1976) · Zbl 0361.53058 · doi:10.4310/jdg/1214433729
[44] Yau, S.T.: Problem Section, Seminar on Differential Geometry. Annals of Mathematics Studies, vol. 102, pp. 669-706. Princeton University Press, Princeton (1982) · Zbl 0471.00020
[45] Ziller, W., Geometry of the Katok examples, Ergod. Theory Dyn. Syst., 3, 1, 135-157 (1983) · Zbl 0559.58027 · doi:10.1017/S0143385700001851
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