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Global properties of tight Reeb flows with applications to Finsler geodesic flows on S2

Published online by Cambridge University Press:  26 July 2012

UMBERTO L. HRYNIEWICZ  and
PEDRO A. S. SALOMÃO
UMBERTO L. HRYNIEWICZ
Affiliation:
Departamento de Matemática Aplicada, Instituto de Matemática, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil. e-mail: [email protected]
PEDRO A. S. SALOMÃO
Affiliation:
Departamento de Matemática, Instituto de Matemática e Estatística, Universidade de São Paulo, São Paulo, Brazil e-mail: [email protected]
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Abstract

We show that if a Finsler metric on S2 with reversibility r has flag curvatures K satisfying (r/(r+1))2 < K ≤ 1, then closed geodesics with specific contact-topological properties cannot exist, in particular there are no closed geodesics with precisely one transverse self-intersection point. This is a special case of a more general phenomenon, and other closed geodesics with many self-intersections are also excluded. We provide examples of Randers type, obtained by suitably modifying the metrics constructed by Katok [21], proving that this pinching condition is sharp. Our methods are borrowed from the theory of pseudo-holomorphic curves in symplectizations. Finally, we study global dynamical aspects of 3-dimensional energy levels C2-close to S3

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 154 , Issue 1 , January 2013 , pp. 1 - 27
Copyright
Copyright © Cambridge Philosophical Society 2012

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References

REFERENCES

[1]Angensent, S.Curve Shortening and the topology of closed geodesics on surfaces. Ann. of Math. (2) 162 (2005), 1185–1239.Google Scholar
[2]Arnol'd, V. I.Topological Invariants of Plane Curves and Caustics. Dean Jacqueline B. Lewis Memorial Lectures presented at Rutgers University, New Brunswick, New Jersey. University Lecture Series, 5 (American Mathematical Society, Providence, RI, 1994).Google Scholar
[3]Bangert, V.On the lengths of closed geodesics on almost round spheres. Math. Z. 191 (1986), no. 4, 549–558.CrossRefGoogle Scholar
[4]Ballmann, W.On the length of closed geodesics on convex surfaces. Invent. Math. 71 (1983), 593–597.CrossRefGoogle Scholar
[5]Ballmann, W., Thorbergsson, G. and Ziller, W.Closed geodesics on positively curved manifolds. Ann. of Math. 116 (1982), 213–247.CrossRefGoogle Scholar
[6]Ballmann, W., Thorbergsson, G. and Ziller, W.Existence of closed geodesics on positively curved manifolds. J. Diffferential Geom. 18 (1983), 221–252.CrossRefGoogle Scholar
[7]Ballmann, W., Thorbergsson, G. and Ziller, W.Some existence theorems for closed geodesics. Comment. Math. Helvetici 58 (1983), 416–432.CrossRefGoogle Scholar
[8]Bao, D. and Robles, C.Ricci and flag curvatures in Finsler geometry. A Sampler of Riemann–Finsler Geometry, 197–259, Math. Sci. Res. Inst. Publ., 50 (Cambridge University Press, 2004).Google Scholar
[9]Birkhoff, G. D.Dynamical Systems. Amer. Math. Soc. Colloq. Publ. 9 (American Mathematical Society, Providence, 1966).Google Scholar
[10]Birkhoff, G. and Rota, G.-C.Ordinary Differential Equations. Fourth edition (John Wiley & Sons, Inc., New York, 1989).Google Scholar
[11]Contreras, G. and Oliveira, F.C 2-densely the 2-sphere has an elliptic closed geodesic. Ergodic Theory Dynam. Systems. 24 (2004), 1395–1423.CrossRefGoogle Scholar
[12]Grifone, J.Structure presque-tangente et connexions I. Ann. Inst. Fourier (Grenoble) 22 (1972), no. 1, 287–334.CrossRefGoogle Scholar
[13]Harris, A. and Paternain, G.Dynamically convex Finsler metrics and J-holomorphic embedding of asymptotic cylinders. Ann. Global Anal. Geom. 34 (2008), no. 2, 115–134.CrossRefGoogle Scholar
[14]Hofer, H.Pseudoholomorphic curves in symplectisations with application to the Weinstein conjecture in dimension three. Invent. Math. 114 (1993), 515–563.CrossRefGoogle Scholar
[15]Hofer, H., Wysocki, K. and Zehnder, E.Properties of pseudoholomorphic curves in symplectisations I: Asymptotics. Ann. Inst. H. Poincaré Anal. Non Linéaire 13 (1996), 337–379.CrossRefGoogle Scholar
[16]Hofer, H., Wysocki, K. and Zehnder, E.A characterization of the tight three sphere. Duke Math. J. 81 (1995), no. 1, 159–226.CrossRefGoogle Scholar
[17]Hofer, H., Wysocki, K. and Zehnder, E.A characterization of the tight three sphere II. Commun. Pure Appl. Anal. 55 (1999), no. 9, 1139–1177.Google Scholar
[18]Hofer, H., Wysocki, K. and Zehnder, E.The dynamics of strictly convex energy surfaces in ℝ4. Ann. of Math. 148 (1998), 197–289.CrossRefGoogle Scholar
[19]Hofer, H., Wysocki, K. and Zehnder, E.Finite energy foliations of tight three-spheres and Hamiltonian dynamics. Ann. of Math. 157 (2003), 125–255.CrossRefGoogle Scholar
[20]Hryniewicz, U.Fast finite-energy planes in symplectizations and applications. Trans. Amer. Math. Soc. 364 (2012), no. 4, 1859–1931.CrossRefGoogle Scholar
[21]Katok, A.Ergodic properties of degenerate integrable Hamiltonian systems. Izv. Akad. Nauk SSSR. 37 (1973) (Russian), 535–571.Google Scholar
[22]Klingenberg, W.Der Indexsatz für geschlossene Geodätsche. Math. Z. 139 (1974), 231–256.CrossRefGoogle Scholar
[23]Poincaré, H.Sur les lignes géodésiques des surfaces convexes. Trans. Amer. Math. Soc. 6 (1905), 237–274.Google Scholar
[24]Rademacher, H.-B.A sphere theorem for non-reversible Finsler metrics. Math. Ann. 328 (2004), 373–387.CrossRefGoogle Scholar
[25]Rademacher, H.-B.Nonreversible Finsler metrics of positive flag curvature. A Sampler of Riemann–Finsler Geometry, 261–302, Math. Sci. Res. Inst. Publ. 50 (Cambridge University Press, 2004).Google Scholar
[26]Rademacher, H.-B.The length of a shortest geodesic loop. C. R. Math. Acad. Sci. Paris 346, 13–14 (2008), 763–765.CrossRefGoogle Scholar
[27]Thorbergsson, G.Non-hyperbolic closed geodesics. Math. Scand. 44 (1979), 135–148.CrossRefGoogle Scholar
[28]Vitório, H. A geometria de curvas fanning e de suas reduções simpléticas. PhD. thesis, Unicamp (2010).Google Scholar