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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: 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
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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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