GEOMETRY AND ENTROPIES IN A FIXED CONFORMAL CLASS ON SURFACES THOMAS BARTHELME´ AND ALENA ERCHENKO Abstract. 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. 1. Introduction If M is a fixed surface, there has been a long history of studying how the geometric or dynamical data (e.g., the Laplace spectrum, systole, entropies or Lyapunov exponents of the geodesic flow) varies when one varies the metric on M, possibly inside a particular class. In [BE17], we studied these questions in a class of metrics that seemed to have been overlooked: the family of non-positively curved metrics within a fixed conformal class. In this article, we prove several conjectures made in [BE17], as well as give a fairly complete, albeit coarse, picture of the geometry of non-positively curved metrics within a fixed conformal class. Since Gromov's famous systolic inequality [Gro83], there has been a lot of interest in upper bounds on the systole (see for instance [Gut10]). In general, there is no positive lower bound on the systole, but we show here that there is an interesting class of metrics with a lower bound on the systole. We prove the following theorem that, in particular, implies Conjecture 1.2 in [BE17]. Theorem A. (see Theorem 2.4 and Corollary 2.5) Let σ be a fixed hyperbolic metric on a closed surface M of negative Euler characteristic. Let A > 0 be fixed. There exist positive constants C1;C2 depending on the topology of M, the metric σ and A such that inf sys(g) ≥ C1 and sup htop(g) ≤ C2; g2[σ]≤ ≤ A g2[σ]A ≤ where [σ]A is the family of smooth non-positively curved Riemannian metrics on M that are conformally equivalent to σ and have total area A. Although the topological entropy of the geodesic flow on M for a non-positively curved metric with fixed total area somehow detects the information about a conformal class, we show that the metric entropy is still flexible in any conformal class, which proves Conjec- ture 1.1 in [BE17]. Theorem B. (Theorem 5.1) Let M be a closed surface of negative Euler characteristic and σ be a hyperbolic metric on M. Suppose A > 0. Then, inf hmetr(g) = 0: < g2[σ]A 1 2 T.BARTHELME´ AND A.ERCHENKO The key ingredient in proving the above theorem is to be able to smooth a conical point of a metric while preserving the conformal class (see Lemma 5.2). Along the way to understanding if there are additional restrictions for entropies in a fixed conformal class, we obtained a better understanding of the geometry of non-positively curved metrics. Recall that a hyperbolic surface (M; σ) can be decomposed into thick parts that have a bounded geometry and thin parts that are homeomorphic to annuli (see [BP92, Chapter D]). We show that the thick-thin decomposition of a hyperbolic surface determines a thick- thin decomposition for non-positively curved metrics that are conformally equivalent to that hyperbolic surface. ≤ Theorem C. (Theorem 4.1) For every thick piece Y of (M; σ), for every g 2 [σ]A, and for every non-trivial non-peripheral piecewise-smooth simple closed curve α in Y , the g- length of the g-geodesic representative of α is comparable to the σ-length of the σ-geodesic representative of α up to a multiplicative constant that depends only on the topology of M, the metric σ and A. In addition, there is a well-known collar lemma for hyperbolic surfaces, i.e., if there exists a short non-trivial simple closed geodesic then the transversal closed geodesics are long. The collar lemma was generalized for Riemannian metrics with a lower curvature bound in [Bus78]. Lemma 2.6 is an analogous result for non-positively curved metrics in a fixed conformal class. Finally,it is natural to study the compactification of a family of metrics. Note that smooth Riemannian metrics of non-positive curvature are metrics with bounded integral curvature (see [Deb16, Section 1.1]) due to the Gauss-Bonnet formula. The curvature measures coming from smooth Riemannian metrics give rise to metrics of bounded integral curvature (see [Tro09, Proposition 5.2]) whose uniform convergence was studied in [Res93, Theorem 7.3.1] and [Tro09, Theorem 6.2]. However, the metrics constructed using Proposition 5.2 in [Tro09] are not necessarily the original Riemannian metrics because curvature measures are invariant under scaling and the constructed conformal coefficient for each metric differs from the initial conformal coefficient of the non-positively curved Riemannian metric (see [Tro09, Theorem 5.1 (e)]). The precompactness of metrics of bounded integral curvature in the uniform metric sense was shown in [Deb16, Main Theorem] under the additional restriction of a uniform lower bound on the length of a systole and uniform upper bounds on area and positive curvature on small geodesic balls. Using TheoremA, we strengthen [Tro09, Theorem 6.2] and [Deb16, Corollary 5] by showing precompactness of the family of non-positively curved Riemannian metrics with a fixed total area in a fixed conformal class. ≤ Theorem D. (Theorem 3.2) The set of metrics [σ]A is precompact in the uniform metric sense (see Definition 3.1) with the limiting metrics having bounded integral curvature. 1.1. Organization of the paper. In Section2, we prove a uniform lower bound on the length of a systole for the family of smooth non-positively curved Riemannian metrics with fixed total area in a fixed conformal class. In Section3, we show precompactness in the uniform metric sense of the considered metrics. We obtain a thick-thin decomposition in Section4. The flexibility of metric entropy is proved in Section5. Some natural open questions are formulated in Section6. GEOMETRY IN A FIXED CONFORMAL CLASS 3 Acknowledgements. We would like to thank Yair Minsky, Federico Rodriguez Hertz, and Dennis Sullivan for useful discussions and questions. We also thank Ian Frankel for pointing out that our result is related to the work of Yurii Reshetnyak. 2. Collar lemma Consider a Riemannian metric g on a closed surface M of Euler characteristic χ(M) < 0. Denote by [g] a family of metrics conformally equivalent to g. Since all of our results apply trivially to any finite cover of M, we always assume M to be orientable. Let γ be a simple closed curve on M and [γ] be a family of simple closed curves isotopic to γ. Denote by lg(·) the g-length of a curve and Ag(·) the g-area of a set. Definition 2.1. The extremal length of [γ] with respect to a Riemannian metric g is 2 0 inf lg0 (γ ) γ02[γ] Eg(γ) = sup 0 : (2.1) g02[g] Ag(M) Notice that Eg(·) depends only on the conformal class of g. Definition 2.2. The modulus Modg(A) of an annulus A on (M; g) is the reciprocal of Eg(γ), where γ is a simple closed curve isotopic to a boundary curve of A. 2 0 inf l 0 (c ) c02[c] g Moreover, Modg(A) = Eg(c) = sup A0 (A) , where c is a path connecting the boundaries g02[g] g of A and [c] is a family of curves that connect the boundaries of A and isotopic to c. In particular, Modg(·) depends only on the conformal class of g. Let γ be a smooth curve that is a boundary of a set S in M. We choose the sign of the geodesic curvature of γ be positive when the acceleration vector points into S. We say that γ is monotonically curved with respect to S if γ is positively (or negatively) curved with respect to S at every point (the terminology from [Min92] and [Raf05]). We denote by κ(γ) the integral of the geodesic curvature of γ. The following result is a (weaker) version of [Raf05, Lemma 3.6], which itself used [Min92, Theorem 4.5], adapted to our context of smooth non-positively cuved Riemannian metrics. Our result is weaker then [Raf05, Lemma 3.6] because we only give a lower bound for the modulus of the annulus, but this is all we will need in order to prove TheoremA Lemma 2.3. Let g be a smooth Riemannian non-positively curved metric on a closed surface M of Euler characteristic χ(M) < 0. Let A be an annulus in (M; g) with monotonically curved equidistant boundary curves γ0 and γ1. Suppose that κ(γ0) ≤ 0. Then, distg(γ0; γ1) Modg(A) = if κ(γ0) = 0 and A is a flat annulus; lg(γ0) 1 distg(γ0; γ1) Modg(A) ≥ ln 1 − 2πχ(M) otherwise, −2πχ(M) lg(γ0) where distg(·; ·) is the distance function on (M; g) Proof. If κ(γ0) = 0 and A is a flat annulus, then this is a classical result (see [Ahl73, Chapter 4]). 4 T.BARTHELME´ AND A.ERCHENKO Consider the level curvesγ ^r := fp 2 Aj distg(p; γ0) = rg. Each curveγ ^r is monotonically curved thanks to the convexity of the distance function distg(·; ·). Moreover, notice that γ^0 = γ0 andγ ^distg(γ0,γ1) = γ1. We define a scaling function f by f(r) := lg(^γ0) . Then, for any r, the curveγ ^ has length, lg(^γr) r in the metric f(r)g, lfg(^γr) = lg(γ0).