Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-10T21:48:05.419Z Has data issue: false hasContentIssue false
  • English
  • Français

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: umberto@labma.ufrj.br
PEDRO A. S. SALOMÃO
Affiliation:
Departamento de Matemática, Instituto de Matemática e Estatística, Universidade de São Paulo, São Paulo, Brazil e-mail: psalomao@ime.usp.br
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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