Geometry and entropies in a fixed conformal class on surfaces

Barthelmé, Thomas; Erchenko, Alena

Annales de l'Institut Fourier, Tome 71 (2021) no. 2, pp. 731-755.

Pour une surface à caractéristique d’Euler négative, nous considérons la classe des métriques à courbure négative, à volume fixé, et conformément équivalent à une métrique fixée. Pour cette classe de métriques, nous montrons que l’entropie métrique du flot géodésique est flexible, mais qu’à contrario, il y a des restrictions sur l’entropie topologique ainsi que la systole. Ce faisant, nous obtenons aussi un lemme du collier, une décomposition « épaisse-fine », et un théorème de précompacité pour cette classe de métrique. Nous étendons aussi certains de nos résultats au classes de métriques où la condition de courbure négative est remplacée par la condition de ne pas avoir de points focaux et d’avoir une certaine borne pour l’intégrale de la partie positive de la courbure de Gauss.

We show the flexibility of the metric entropy and obtain additional restrictions on the topological entropy of geodesic flow on closed surfaces of negative Euler characteristic with smooth non-positively curved Riemannian metrics with fixed total area in a fixed conformal class. Moreover, we obtain a collar lemma, a thick-thin decomposition, and precompactness for the considered class of metrics. Also, we extend some of the results to metrics of fixed total area in a fixed conformal class with no focal points and with some integral bounds on the positive part of the Gaussian curvature.

Classification : 37D40, 53C20
Keywords: conformal class, topological entropy, metric entropy
Mot clés : classe conforme, entropie topologique, entropie métrique

[1] Ahlfors, Lars V. Conformal invariants: topics in geometric function theory, McGraw-Hill Series in Higher Mathematics, McGraw-Hill, 1973, ix+157 pages | Zbl

[2] Barthelmé, Thomas; Erchenko, Alena Flexibility of geometric and dynamical data in fixed conformal classes, Indiana Univ. Math. J., Volume 69 (2020) no. 2, pp. 513-540 | MR | Zbl

[3] Benedetti, Riccardo; Petronio, Carlo Lectures on hyperbolic geometry, Universitext, Springer, 1992, xiv+330 pages | DOI

[4] Burago, Yuri D.; Zalgaller, Viktor A. Geometric inequalities, Grundlehren der Mathematischen Wissenschaften, 285, Springer, 1988, xiv+331 pages | DOI | MR

[5] Buser, Peter The collar theorem and examples, Manuscr. Math., Volume 25 (1978) no. 4, pp. 349-357 | DOI | MR | Zbl

[6] Chavel, Isaac Riemannian geometry—a modern introduction, Cambridge Tracts in Mathematics, 108, Cambridge University Press, 1993, xii+386 pages | MR

[7] Debin, Clément A compactness theorem for surfaces with Bounded Integral Curvature, J. Inst. Math. Jussieu (2018), pp. 1-49 | MR | Zbl

[8] Erchenko, Alena; Katok, Anatole Flexibility of entropies for surfaces of negative curvature, Isr. J. Math., Volume 232 (2019) no. 2, pp. 631-676 | DOI | MR | Zbl

[9] Gromov, Mikhael Filling Riemannian manifolds, J. Differ. Geom., Volume 18 (1983) no. 1, pp. 1-147 | MR | Zbl

[10] Guth, Larry Metaphors in systolic geometry, Proceedings of the International Congress of Mathematicians. Volume II, Hindustan Book Agency, New Delhi (2010), pp. 745-768 | MR | Zbl

[11] Izmestiev, Ivan A simple proof of an isoperimetric inequality for Euclidean and hyperbolic cone-surfaces, Differ. Geom. Appl., Volume 43 (2015), pp. 95-101 | DOI | MR | Zbl

[12] Katok, Anatole Entropy and closed geodesics, Ergodic Theory Dyn. Syst., Volume 2 (1982) no. 3-4, pp. 339-365 | DOI | MR | Zbl

[13] Kerckhoff, Steven The asymptotic geometry of Teichmüller space, Topology, Volume 19 (1980) no. 1, pp. 23-41 | DOI | MR | Zbl

[14] Ledrappier, François Propriété de Poisson et courbure négative, C. R. Math. Acad. Sci. Paris, Volume 305 (1987) no. 5, pp. 191-194 | Zbl

[15] Manning, Anthony Topological entropy for geodesic flows, Ann. Math., Volume 110 (1979) no. 3, pp. 567-573 | DOI | MR | Zbl

[16] Minsky, Yair Harmonic maps, length, and energy in Teichmüller space, J. Differ. Geom., Volume 35 (1992) no. 1, pp. 151-217 | Zbl

[17] Minsky, Yair Teichmüller geodesics and ends of hyperbolic $3$-manifolds, Topology, Volume 32 (1993) no. 3, pp. 625-647 | DOI | Zbl

[18] O’Sullivan, John J. Manifolds without conjugate points, Math. Ann., Volume 210 (1974), pp. 295-311 | DOI | MR | Zbl

[19] Rafi, Kasra A characterization of short curves of a Teichmüller geodesic, Geom. Topol., Volume 9 (2005), pp. 179-202 | DOI | Zbl

[20] Rafi, Kasra Thick-thin decomposition for quadratic differentials, Math. Res. Lett., Volume 14 (2007) no. 2, pp. 333-341 | DOI | MR | Zbl

[21] Ramos, Daniel Smoothening cone points with Ricci flow, Bull. Soc. Math. Fr., Volume 143 (2015) no. 4, pp. 619-633 | DOI | MR | Zbl

[22] Reshetnyak, Yuri Two-dimensional manifolds of bounded curvature, Geometry, IV (Encyclopaedia of Mathematical Sciences), Volume 70, Springer, 1993, pp. 3-163 | DOI | MR

[23] Ruelle, David An inequality for the entropy of differentiable maps, Bol. Soc. Bras. Mat., Volume 9 (1978) no. 1, pp. 83-87 | DOI | MR | Zbl

[24] Sabau, Sorin V.; Tanaka, Minoru The cut locus and distance function from a closed subset of a Finsler manifold, Houston J. Math., Volume 42 (2016) no. 4, pp. 1157-1197 | MR | Zbl

[25] Sabourau, Stéphane Entropy and systoles on surfaces, Ergodic Theory Dyn. Syst., Volume 26 (2006) no. 5, pp. 1653-1669 | DOI | MR | Zbl

[26] Troyanov, Marc Les surfaces euclidiennes à singularités coniques, Enseign. Math., Volume 32 (1986) no. 1-2, pp. 79-94 | Zbl

[27] Troyanov, Marc Prescribing curvature on compact surfaces with conical singularities, Trans. Am. Math. Soc., Volume 324 (1991) no. 2, pp. 793-821 | DOI | MR | Zbl

[28] Troyanov, Marc Les surfaces à courbure intégrale bornée au sens d’Alexandrov, Géométrie discrète, algorithmique, différentielle et arithmétique (SMF Journée Annuelle), Volume 2009, Société Mathématique de France, 2009, pp. 1-18 | Zbl