Abstract
Given an arbitrary Riemannian metric on a closed surface, we consider length-minimizing geodesics in the universal cover. Morse and Hedlund proved that such minimal geodesics lie in bounded distance of geodesics of a Riemannian metric of constant curvature. Knieper asked when two minimal geodesics in bounded distance of a single constant-curvature geodesic can intersect. In this paper we prove an asymptotic property of minimal rays, showing in particular that intersecting minimal geodesics as above can only occur as heteroclinic connections between pairs of homotopic closed minimal geodesics. A further application characterizes the boundary at infinity of the universal cover defined by Busemann functions. A third application shows that flat strips in the universal cover of a nonpositively curved surface are foliated by lifts of closed geodesics of a single homotopy class.
Similar content being viewed by others
References
W. Ballmann, Lectures on Spaces of Nonpositive Curvature. With an appendix by Misha Brin, DMV Seminar, Vol. 25, Birkhäuser Verlag, Basel, 1995.
V. Bangert, Mather sets for twist maps and geodesics on tori, in Dynamics Reported, Vol. 1, Wiley, Chichester, 1988, pp. 1–56.
D. D.-W. Bao, S. S. Chern and Z. Shen, An Introduction to Riemann–Finsler Geometry, Graduate Texts in Mathematics, Vol. 200, Springer Verlag, New York, 2000.
M. L. Bialy and L. V. Polterovich, Geodesic flows on the two-dimensional torus and phase transitions ‘commensurability-noncommensurability’ (English translation), Functional Analysis and its Applications 20 (1986), 260–266.
M. R. Bridson and A. Haefliger, Metric Spaces of Non-positive Curvature, Grundlehren der Mathematischen Wissenschaften, Vol. 319, Springer Verlag, Berlin, 1999.
G. Contreras, R. Iturriaga, G. P. Paternain and M. Paternain, Lagrangian graphs, minimizing measures and Ma˜né’s critical values, Geometric and Functional Analysis 8 (1998), 788–809.
Y. Coudéne and B. Schapira, Generic measures for geodesic flows in non-positively curved manifolds, Journal de l’École Polythechnique 1 (2014), 387–408.
M. J. Dias Carneiro and R. O. Ruggiero, On Birkhoff Theorems for Lagrangian invariant tori with closed orbits, Manuscripta Mathematica 119 (2006), 411–432.
P. Eberlein and B. O’Neill, Visibility manifolds, Pacific Journal of Mathematics 46 (1973), 45–109.
A. Fathi, Weak KAM Theorem in Lagrangian Dynamics, Cambridge University Press, to appear.
E. Glasmachers, G. Knieper, C. Ogouyandjou and J. P. Schröder, Topological entropy of minimal geodesics and volume growth on surfaces, Journal of Modern Dynamics 8 (2014), 75–91.
M. Gromov, Hyperbolic manifolds, groups and actions, in Riemann Surfaces and Related Topics: Proceedings of the 1978 Stony Brook Conference, Annals of Mathematics Studies, Vol. 97, Princeton University Press, Princeton, NJ, 1981, pp. 183–213.
G. A. Hedlund, Geodesics on a two-dimensional Riemannian manifold with periodic coefficients, Annals of Mathematics 33 (1932), 719–739.
E. Hopf, Closed surfaces without conjugate points, Proceedings of the National Academy of Sciences of the United States of America 34 (1948), 47–51.
W. Klingenberg, Geodätischer Fluß auf Mannigfaltigkeiten vom hyperbolischen Typ, Inventiones Mathematicae 14 (1971), 63–82.
D. Massart and A. Sorrentino, Differentiability of Mather’s average action and integrability on closed surfaces, Nonlinearity 24 (2011), 1777–1793.
J. N. Mather, Action minimizing measures for positive definite Lagrangian systems, Mathematische Zeitschrift 207 (1991), 169–207.
H. M. Morse, A fundamental class of geodesics on any closed surface of genus greater than one, Transactions of the American Mathematical Society 26 (1924), 25–60.
J. P. Schröder, Global minimizers for Tonelli Lagrangians on the 2-torus, Journal of Topology and Analysis 7 (2015), 261–291.
J. P. Schröder, Ergodicity and topological entropy of geodesic flows on surfaces, Journal of Modern Dynamics 9 (2015), 147–167.
J. P. Schröder, Invariant tori and topological entropy in Tonelli Lagrangian systems on the 2-torus, Ergodic Theory and Dynamical Systems 36 (2016), 1989–2014.
J. P. Schröder, Generic uniqueness of shortest closed geodesics, Calculus of Variations and Partial Differential Equations 55 (2016), Article 55.
E. M. Zaustinsky, Extremals on compact E-surfaces, Transactions of the American Mathematical Society 102 (1962), 433–445.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Schröder, J.P. Minimal rays on closed surfaces. Isr. J. Math. 217, 197–229 (2017). https://doi.org/10.1007/s11856-017-1443-9
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11856-017-1443-9