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Hopf rigidity for convex billiards on the hemisphere and hyperbolic plane

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  • This paper deals with Hopf type rigidity for convex billiards on surfaces of constant curvature. I prove that the only convex billiard without conjugate points on the hyperbolic plane or on the hemisphere is a circular billiard.
    Mathematics Subject Classification: Primary: 37J35, 37J50; Secondary: 37E40.

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  • [1]

    M.-C. Arnaud, A particular minimization property implies $C^0$ -integrability, J. Differential Equations, 250 (2011), 2389-2401.doi: 10.1016/j.jde.2010.12.002.

    [2]

    M. Bialy, Convex billiards and a theorem by E. Hopf, Math. Z., 214 (1993), 147-154.doi: 10.1007/BF02572397.

    [3]

    M. Bialy, Maximizing orbits for higher dimensional convex billiards, Journal of Modern Dynamics, 3 (2009), 51-59.doi: 10.3934/jmd.2009.3.51.

    [4]

    V. Blumen, K. Y. Kim, J. Nance and V. Zharnitsky, Three-period orbits in billiards on the surfaces of constant curvature Int. Math. Res. Notices, 2012 (2012), 5014-5024.

    [5]

    Yu. Burago and V. Zalgaller, "Geometric Inequalities," Translated from the Russian by A. B. Sosinskiĭ, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 285, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1988.

    [6]

    D. Burago and S. Ivanov, Riemannian tori without conjugate points are flat, Geom. Funct. Anal., 4 (1994), 259-269.doi: 10.1007/BF01896241.

    [7]

    B. Gutkin, U. Smilansky and E. Gutkin, Hyperbolic billiards on surfaces of constant curvature, Comm. Math. Phys., 208 (1999), 65-90.doi: 10.1007/s002200050748.

    [8]

    E. Gutkin and S. Tabachnikov, Billiards in Finsler and Minkowski geometries, J. Geom. Phys., 40 (2002), 277-301.doi: 10.1016/S0393-0440(01)00039-0.

    [9]

    E. Gutkin, Billiard dynamics: A survey with the emphasis on open problems, Regul. Chaotic Dyn., 8 (2003), 1-13.doi: 10.1070/RD2003v008n01ABEH000222.

    [10]

    S. Elaydi, "An Introduction to Difference Equations," Third edition, Undergraduate Texts in Mathematics, Springer, New York, 2005.

    [11]

    E. Hopf, Closed surfaces without conjugate points, Proc. Nat. Acad. Sci. U. S. A., 34 (1948), 47-51.

    [12]

    J. Heber, On the geodesic flow of tori without conjugate points, Math. Z., 216 (1994), 209-216.doi: 10.1007/BF02572318.

    [13]

    N. Innami, Integral formulas for polyhedral and spherical billiards, J. Math. Soc. Japan, 50 (1998), 339-357.doi: 10.2969/jmsj/05020339.

    [14]

    V. Kaloshin and A. SorrentinoOn conjugacy of convex billiards, preprint, arXiv:1203.1274.

    [15]

    A. Knauf, Closed orbits and converse KAM theory, Nonlinearity, 3 (1990), 961-973.

    [16]

    R. MacKay, J. Meiss and J. Strark, Converse KAM theory for symplectic twist maps, Nonlinearity, 2 (1989), 555-570.

    [17]

    A. Veselov, Confocal surfaces and integrable billiards on the sphere and in the Lobachevsky space, J. Geom. Phys., 7 (1990), 81-107.doi: 10.1016/0393-0440(90)90021-T.

    [18]

    M. Wojtkowski, Two applications of Jacobi fields to the billiard ball problem, J. Differential Geom., 40 (1994), 155-164.

    [19]

    S. Tabachnikov, Billiards, Panor. Synth., 1 (1995), vi+142 pp.

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