Geometry and Entropies in a Fixed Conformal Class on Surfaces

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, g∈[σ]≤ ≤ A g∈[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. < g∈[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 ∈ [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 ) γ0∈[γ] Eg(γ) = sup 0 . (2.1) g0∈[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 ) c0∈[c] g Moreover, Modg(A) = Eg(c) = sup A0 (A) , where c is a path connecting the boundaries g0∈[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 ˆr := {p ∈ A| distg(p, γ0) = r}. Each curveγ ˆr is monotonically curved thanks to the convexity of the distance function distg(·, ·). Moreover, notice that

γˆ0 = γ0 andγ ˆdistg(γ0,γ1) = γ1. We deﬁne a scaling function f by f(r) := lg(ˆγ0) . Then, for any r, the curveγ ˆ has length, lg(ˆγr) r in the metric f(r)g, lfg(ˆγr) = lg(γ0). Let Ar be the annulus in A that is bounded by γ0 andγ ˆr. In order to bound from below the modulus of A, we will use that it is the extremal length of paths connecting the two boundaries. Consider c a path from γ0 to γ1. Then, its fg-length satisﬁes

dist (γ ,γ ) dist (γ ,γ ) dist (γ ,γ ) Zg 0 1 Zg 0 1 Zg 0 1 lg(γ0) 1 lfg(c) ≥ distfg(γ0, γ1) = f(r)dr = dr = lg(γ0) dr. lg(ˆγr) lg(ˆγr) 0 0 0 Moreover,

distg(γ0,γ1) distg(γ0,γ1) distg(γ0,γ1) Z Z 2 Z 2 lg(γ0) 2 1 volfg(A) = f (r)lg(ˆγr)dr = dr = lg(γ0) dr. lg(ˆγr) lg(ˆγr) 0 0 0

Let Kg(·) denote the Gaussian curvature function on (M, g). Applying Gauss-Bonnet Theorem to (M, g) and (A , g) gives that R K (x)dA (x) = 2πχ(M) and R K dA + r M g g Ar g g κ(ˆγr) − κ(γ0) = 0. Now, our assumptions are that χ(M) < 0, κ(γ0) ≤ 0, and Kg(x) ≤ 0 for any x ∈ (M, g). Thus we have κ(ˆγr) ≤ −2πχ(M). Combining the previous inequality d with the fact that dr lg(ˆγr) = κ(ˆγr), where κ(ˆγr) is the integral of the geodesic curvature, we obtain lg(ˆγr) ≤ −2πχ(M)r + lg(γ0). Therefore, using Deﬁnition 2.2, we have

dist (γ ,γ ) dist (γ ,γ ) 2 Zg 0 1 Zg 0 1 distfg(γ0, γ1) 1 1 Modg(A) ≥ = dr ≥ dr volfg(A) lg(ˆγr) lg(γ0) − 2πχ(M)r 0 0 1 dist (γ , γ ) = ln 1 − 2πχ(M) g 0 1 . −2πχ(M) lg(γ0) Theorem 2.4. Let M be a closed surface of Euler characteristic χ(M) < 0 and σ be a hyperbolic metric on M. Suppose A > 0. Then, there exists a positive constant C = C(σ, A) such that inf sys(g) ≥ C, ≤ g∈[σ]A ≤ where [σ]A is a family of smooth non-positively curved metric conformally equivalent to σ with total area A and sys(g) is the length of the shortest simple nontrivial closed geodesic for q A the metric g. In particular, we have C = R , where E = E(σ) = sup Modσ(A), A− annulus ⊂M 2 1 e2πE(χ2(M)−3χ(M)) − 1 R = π(χ2(M) − χ(M))Rˆ2 + (1 − πχ(M))Rˆ + , and Rˆ = . π π −πχ(M) GEOMETRY IN A FIXED CONFORMAL CLASS 5

Theorem 1.2 in [Sab06] says that there exists a constant C > 0 such that for every Riemannian metric g on M we have

sys(g)hvol(g) ≤ C, where hvol(·) is the volume entropy on (M, g). Moreover, the topological entropy coincides with the volume entropy for non-positively curved metrics [Man79, Theorem 2]. Therefore, we obtain the following corollary of Theorem 2.4, which, in particular, proves Conjecture 1.2 of [BE17]. Corollary 2.5. Let M be a closed surface of the negative Euler characteristic and σ be a hyperbolic metric on M. Suppose A > 0. Then, there exists a positive constant B = B(σ, A) such that sup htop(g) ≤ B, ≤ g∈[σ]A where htop(g) is the topological entropy of the geodesic ﬂow on M with respect to the metric g. In order to prove Theorem 2.4, we will follow the proof of [Raf05, Lemma 4.1] while adapting it to our setting. Proof of Theorem 2.4. Let g be a smooth non-positively curved Riemannian metric on M wit total area A and conformally equivalent to σ. Denote by γ the shortest simple closed nontrivial geodesic for g. Let Nr be the open r-neighborhood of γ in (M, g). Then, we deﬁne Zr to be the union of Nr and all components of M \ Nr that are disks. Let κ(∂Zr) be the integral of the geodesic curvatures of the boundary components of Zr. We denote by Kg(·) the Gaussian curvature function of (M, g). By the Gauss-Bonnet theorem applied to (M, g) and (Zr, g), we have Z Kg(x)dAg(x) + κ(∂Zr) = 2πχ(Zr) and (2.2) Zr Z Kg(x)dAg(x) = 2πχ(M), respectively, M where χ(·) is the Euler characteristic function. By (2.2), we have

κ(∂Zr) ≤ −2πχ(M) = K for any r (2.3) as χ(Zr) ≤ 0 and Kg(x) ≤ 0 for any x ∈ (M, g). We recall that functions lg(∂Zr) and Ag(Zr) are differentiable functions of r everywhere except for finitely many r, where we add a disk. For those r where the functions are d d differentiable we have dr lg(∂Zr) = κ(∂Zr) and dr Ag(Zr) = lg(∂Zr). Define Ir to be the set of all indexes u such that Zru = Nru ∪ Du where Du is the union of disjoint disks and ru ≤ r. Let cu = lg(∂Du), i.e., it is the g-length of the boundary of Du. As a result, we obtain that r Z X lg(∂Zr) − lg(∂Z0) = κ(∂Zτ )dτ − cu, (2.4) 0 u∈Ir r Z X Ag(Zr) − Ag(Z0) = lg(∂Zτ )dτ + Ag(Du). 0 u∈Ir 6 T.BARTHELME´ AND A.ERCHENKO

Furthermore, by the isoperimetric inequality (see [Izm15]), we have

c2 A (D ) ≤ u . (2.5) g u 4π

Therefore, combining (2.3), (2.4),(2.5) and the facts that lg(∂Z0) = 2lg(γ) and Ag(Z0) = 0, the following inequalities hold:

lg(∂Zr) ≤ Kr + 2lg(γ), (2.6) X cu ≤ Kr + 2lg(γ), (2.7)

u∈Ir and r ! Z 1 X A (Z ) ≤ (Kτ + 2l (γ))dτ + c2 g r g 4π u 0 u∈Ir !2 Kr2 1 X ≤ + 2l (γ)r + c (2.8) 2 g 4π u u∈Ir Kr2 1 ≤ + 2l (γ)r + (Kr + 2l (γ))2 . 2 g 4π g s Let r0 = 0 and {ri}i=1 be the increasing sequence of values of r where the topology of Zr changes. Notice that s ≤ 2 − χ(M) because g has non-positive curvature and Zs = M. By s the deﬁnition of {ri}i=0, Zri+1 \Zri is a union of annuli with monotonically curved equidistant

boundary curves for every i = 0, . . . , s − 1. Moreover, for each annuli in Zri+1 \ Zri we have that the distance between its boundaries is ri+1 − ri (by construction) and the length of the shorter boundary is at most Kri + 2lg(γ) (see (2.6)). By Lemma 2.3 and the deﬁnition of the constant E, we have eEK − 1 r − r ≤ (Kr + 2l (γ)) i+1 i K i g Therefore, for any i = 0, . . . , s − 1, we have

ri+1 ≤ P ri + Qlg(γ), (2.9)

EK eEK −1 where P = e and Q = 2 K . By applying (2.9) several times and having r0 = 0, we obtain

rs ≤ P rs−1 + Qlg(γ) ≤ P (P rs−1 + Qlg(γ)) + Qlg(γ) ≤ ... s X P s+1 − 1 ≤ P s+1r + Ql (γ) P i ≤ Q l (γ) (2.10) 0 g P − 1 g i=0

Since Zs = M, by (2.8) and (2.10), we obtain Kr2 1 A = A (Z ) ≤ s + 2l (γ)r + (Kr + 2l (γ))2 ≤ Rl2(γ), g s 2 g s 4π s g g GEOMETRY IN A FIXED CONFORMAL CLASS 7

where 2 1 R = π(χ2(M) − χ(M))Rˆ2 + (1 − πχ(M))Rˆ + and π π e−2πEχ(M) − 1 e−2πEχ(M)(s+1) − 1 e−2πEχ(M)(s+1) − 1 Rˆ = 2 · = . −2πχ(M) e−2πEχ(M) − 1 −πχ(M) Therefore, we have r A sys(g) = l (γ) ≥ . g R Using the fact that s ≤ 2 − χ(M), we prove the theorem. Applying the adaptations of the proof of Theorem 2.4 in the proof of [Raf05, Lemma 4.1], we also can obtain the following collar lemma. Lemma 2.6 (Collar lemma). Consider a closed surface M of negative Euler characteristic. For every L > 0, there exists a constant DL such that the following holds. Let α and β be any two simple closed curves in M that intersect non-trivially. Let g be a smooth non-positively curved Riemannian metric on M that is conformally equivalent to the hyperbolic metric σ. If lσ(βσ) ≤ L, then we have DLlg(αg) ≥ lg(βg),

where βσ is the σ-geodesic representative of β and αg and βg are the g-geodesic representatives of α and β, respectively.

≤ 3. Compactification of [σ]A As before, M is closed orientable surface of negative Euler characteristic and σ is a hyper- ≤ bolic metric on M. Let A be a positive number. Denote by [σ]A the set of smooth negatively curved Riemannian metrics with the total area A that are conformally equivalent to σ.

Deﬁnition 3.1. A sequence of metrics {gk} converges to a metric g on M in the uniform ∗ metric if there are diﬀeomorphisms φk : M → M with φk(gk) → g as k → ∞ uniformly on M. ≤ Theorem 3.2. The set of metrics [σ]A is precompact in the uniform metric sense. Moreover, if a metric g belongs to the limiting set, then g is a metric with bounded integral curvature (see [Deb16, Section 1.1]).

≤ Proof. Theorem 2.4 and [Deb16, Corollary 5] show that the set of metrics [σ]A is precompact in the uniform metric sense and the limiting metrics have bounded integral curvature. 4. Thick-thin decomposition A hyperbolic surface (M, σ) can be decomposed into thick and thin parts (see [BP92, Chap- ter D]). The thin part has a simple topology because the components of it are homeomorphic to annuli. The thick part has a bounded geometry with the diameter and the injectivity radius of a component of the thick part being bounded below and above by constant depending only on the topology of M. In this section we show that a thick component equipped with non-positively curved metric in the conformal class of σ of a ﬁxed total area has a geometry comparable with σ-geometry for that piece. The following theorem is the analogue of [Raf07, Theorem 1] in our setting. 8 T.BARTHELME´ AND A.ERCHENKO

Theorem 4.1. Let M be a closed surface of negative Euler characteristic and σ be a hyperbolic metric on M. Suppose A > 0. Denote by Y a component of the thick part of (M, σ). Then, there exist positive constants C1,C2 that depend only on σ, A, and the topology of M such that for any non-trivial non-peripheral piecewise-smooth simple closed curve α in Y and any smooth non-positively curved metric g conformally equivalent to σ with total area A, we have C1lσ(ασ) ≤ lg(αg) ≤ C2lσ(ασ), (4.1)

where αg is the g-geodesic representative of α. We will need the following lemma for the proof. Lemma 4.2. (version of [Raf07, Lemma 5] in a ﬁxed conformal class) The setting is as in Theorem 4.1. Let α and β be two non-trivial non-peripheral piecewise-smooth simple closed curves in Y . Then, there exists a positive constant D = D(σ, A, topology of M) such that for any smooth non-positively curved metric g conformally equivalent to σ with total area A, we have lg(α)lg(β) ≥ D i(α, β), where i(·, ·) is the intersection number. Proof. The lemma follows from the proof of Lemma 5 in [Raf07] and the fact that for any smooth non-positively curved metric conformally equivalent to σ with total area A the g- size of Y (see the introduction of [Raf07, Section 3]) is bounded below by C from our Theorem 2.4. Proof of Theorem 4.1. Let α be a non-trivial non-peripheral piecewise-smooth simple closed curve in Y . It is straightforward from [BE17, Theorem A] that there exists a constant C2 = C2(σ, A) such that lg(ασ) ≤ C2lσ(ασ). (4.2)

Combining (4.2) with the fact that lg(αg) ≤ lg(ασ) , we have the right inequality in (4.1). Now we show the left inequality in (4.1). Let µ be a short marking of Y , which is a collection of the following curves: all non-trivial simple closed σ-geodesics in the σ-shortest pants decomposition of Y and a collection of transverse non-trivial non-peripheral simple closed curves for each of these curves with the P shortest σ-length. Let Lσ(µ) = lσ(µ) be the σ-length of µ which depends only on σ and β∈µ topology of M. Then, by Lemma 4.2 and [BE17, Theorem A], we obtain that X X X D i(αg, β) ≤ lg(αg)lg(β) ≤ lg(αg) C2lσ(β) = C2Lσ(µ)lg(αg). β∈µ β∈µ β∈µ Finally, we have D X D X D lg(αg) ≥ i(αg, β) = i(ασ, β) ≥ D2lσ(ασ), C2Lσ(µ) C2Lσ(µ) C2Lσ(µ) β∈µ β∈µ

where in the last inequality we used that there exists a positive constant D2 that depends on P σ and the topology of M such that i(ασ, β) ≥ D2lσ(ασ) (see the proof of [Min93, Lemma β∈µ DD2 4.7]). As a result, we get the left inequality in (4.1) with C1 = . C2Lσ(µ) GEOMETRY IN A FIXED CONFORMAL CLASS 9

5. Flexibility of metric entropy In this section we prove Conjecture 1.1 in [BE17]. Theorem 5.1. Let M be a closed orientable surface of negative Euler characteristic and σ be a hyperbolic metric on M. Suppose A > 0. Then,

inf hmetr(g) = 0, < g∈[σ]A where hmetr(g) is the metric entropy with respect to the Liouville measure of the geodesic ﬂow < on (M, g) and [σA ] as in Corollary 2.5. To prove Theorem 5.1, we will need the following lemma. Lemma 5.2. (analogue of [Ram15, Lemma 1] for negatively curved metrics) 2a0(z)+β ln |z| 2 Denote by D the unit disk. Let g0 = e |dz| be a cone metric on the punctured

disk D \{0} with non-positive curvature Kg0 (·), where β > 0 and a0(·) is a smooth function 2uk 2 on D. There exists a decreasing sequence of smooth metrics gk = e |dz| on D such that: 1 (i) gk = g0 on D \ D 1 , where D 1 is a disk of radius centered at 0; k k k (ii) uk ≥ u0 on D \{0}; (iii) inf uk → −∞ as k → +∞; D 1 k

(iv) The Gaussian curvature function Kgk (·) on (D, gk) satisfies Kgk (z) ≤ 0 for any z ∈ D. Remark 5.3. Lemma 5.2 can be of independent interest. In particular, it can be used to define the Ricci flow in the spirit of [Ram15, Theorem 3.1] on surfaces of non-positive curvature everywhere except finitely many points where it has conical singularities with angles larger than 2π. This Ricci flow will smoothen conical points while preserving non- positive curvature. Proof of Lemma 5.2. The proof follows the ideas in [Ram15, Lemma 1]. Let (r, θ) be polar coordinates on D. Consider the conformal factor

u0(r, θ) = a0(r, θ) + β ln r of the metric√ g0. We notice that u0(r, θ) tends to −∞ as r → 0. For each natural number 2 1+ 1+β 2 1 1 k > we deﬁne vk(r) = Ck −ln(1−r ), where Ck = ln(1− 2 )−1+β ln +min a0(r, θ). β k k D In particular,√ vk(0) = Ck → −∞ as k → +∞ and u0(r, θ) − vk(r) ≥ 1 for any θ and 2 1 1+β −1 2v 2 r ∈ [ , ]. Moreover, the metric e k |dz| on D 1 has constant negative curvature k β k −4e−2Ck → −∞ as k → +∞. Choose a smooth function ψ : R → R such that (1) ψ(s) = −s for s ≤ −1; (2) ψ(s) = 0 for s ≥ 1; (3) −1 ≤ ψ0(s) ≤ 0 and ψ00(s) ≥ 0 for any s. Deﬁne a smooth function  1 p1 + β2 − 1  ψ(u0 − vk) + u0 if r ∈ [0, + ], uk = 2k 2β (5.1)  u0 otherwise. 10 T.BARTHELME´ AND A.ERCHENKO

In particular (ii) holds because ψ(s) ≥ 0 for every s ∈ R. The function uk is smooth√ because u0 is smooth outside of any neighborhood of r = 0, 1 1 1+β2−1 uk = u0 for r ∈ [ k , 2k + 2β ], and uk = vk in some neighborhood of r = 0. In particular, (i) and (iii) in Lemma 5.2 holds. Moreover, we have 0 0 (a) Let D be a subset of D such that u0(z) ≤ vk(z) − 1 for z ∈ D . Then, uk = vk and −2Ck 0 Kgk = −4e < 0 on D . 00 00 (b) Let D be a subset of D such that u0(z) ≥ vk(z) + 1 for z ∈ D . Then, uk = u0 and 00 Kgk ≤ 0 on D . 000 000 (c) Let D be a subset of D such that vk(z) − 1 < u0(z) < vk(z) + 1 for z ∈ D . We 00 need to check that Kgk (z) ≤ 0 for z ∈ D . −2uk Recall that Kgk = −e ∆uk. Therefore, Kgk ≤ 0 if and only if ∆uk ≥ 0. In 000 particular, ∆u0 ≥ 0 on D as Kg0 ≤ 0 on D \{0} Using the conditions on ψ, we have the following on D000: 00 2 0 ∆uk = ψ (u0 − vk)|∇(u0 − vk)| + ψ (u0 − vk)∆(u0 − vk) + ∆u0 4 ≥ −ψ0(u − v )∆v = −ψ0(u − v ) ≥ 0. 0 k k 0 k (1 − r2)2 Therefore, (iv) in Lemma 5.2 holds. Proof of Theorem 5.1. Pick a point p on M. Then, by [Tro86, Théoréme,Section 5], there exists a unique metric g of zero curvature everywhere except the point p where it has the conical singularity with angle α = 2π(1 − χ(M)) which has the total area A and is conformally equivalent to σ. In particular, p admits an open neighborhood U and there exists a diffeomorphism from U\{p} to D \{0} such that the metric g on the coordinates of D \{0} have the following expression g = (β + 1)2r2β|dz|2, α where D is the unit disk and β = 2π − 1 > 0. 2uk 2 Denote by gk = e |dz| a family of smooth metrics that we obtain in Lemma 5.2 applied 1 1 to the metric g. The g-radius of the disk D 1 of radius r = centered at 0 is equal to β+1 k k k 1 and has g-area π(β + 1) 2β+2 . In particular, the g-radius and g-area of D 1 tends to 0 as k k k → +∞. Using the notations in the proof of Lemma 5.2, we have 1 1 u = u (r) = ln(β + 1) + β ln r and v (r) = ln(1 − ) − 1 + β ln + ln(β + 1) − ln(1 − r2). 0 0 k k2 k

1 1 q β In particular, u0( k ) − vk( k ) = 1 and u0(r) − vk(r) increases for r ∈ (0, β+2 ) and decreases q β 1 −2β−1 1 β−1 for r ∈ ( β+2 , 1). Moreover, u0(r) − vk(r) ≤ −1 and uk = vk if r ∈ [0, k e (1 − k2 ) ]. 2+2β Therefore, Kg (z) −Ck if z ∈ D 1 , where C is positive constant independent of k. k & k Finally, applying the argument in [EK17, Section 3.3], we obtain

hmetr(gk) → 0 as k → ∞.

In particular, inf hmetr(g) = 0. ≤ g∈[σ]A GEOMETRY IN A FIXED CONFORMAL CLASS 11

Let ε0 > 0 be a suﬃciently small number. Then, by [Tro91, Theorem A], for any 0 ≤ ε < ε0 there exists a metric gε of constant curvature −ε everywhere except a point where it has the conical singularity with angle larger than 2π which has the total area A and is conformally equivalent to σ. Following the same argument as above by starting with metric gε, we obtain

inf hmetr(g) = 0. < g∈[σ]A 6. Further questions In this section, M is a closed orientable surface of negative Euler characteristic χ(M) and < σ is a hyperbolic metric on M. Let A be a positive number. Denote by [σ]A the set of smooth negatively curved Riemannian metrics with the total area A that are conformally equivalent to σ. 6.1. Possible values of entropies in a ﬁxed conformal class. By [Kat82, Theorem B], we know that for any smooth negatively curved Riemannian metric g on M which is not a metric of constant curvature, we have the following inequalities for the metric entropy hmetr(g) with respect to the Liouville measure and the topological entropy htop(g) of the geodesic ﬂow on (M, g)

1 −2πχ(M) 2 0 < h (g) < < h (g). (6.1) metr A top In Theorem 5.1 and Corollary 2.5, we showed that in a fixed conformal class metric entropy can be arbitrary close to 0 while there is an additional constraint on the topological entropy < when we consider metrics from [σ]A. To be more precise, there exists an upper bound for the topological entropy in the fixed conformal class which depends on the conformal class and A. With the discover of a new constraint in the fixed conformal class and the knowledge of the flexibility of the metric entropy, it is natural to try to understand the possible pairs < (hmetr(g), htop(g)) where g ∈ [σ]A. Question 6.1. What is the graph of a function top < Hσ (x) := sup{htop(g) | g ∈ [σ]A, hmetr(g) = x} 1 −2πχ(M) 2 where x ∈ 0, A ?

While it seems hard to answer Question 6.1, it will be a good first step to answer the following questions. 1 −2πχ(M) 2 < ≤ Question 6.2. For any x ∈ 0, A , does there exists g ∈ [σ]A (or g ∈ [σ]A) such that top hmetr(g) = x and htop(g) = Hσ (x)? top Question 6.3. If lim Hσ (x) exists, what is its value in terms of σ? x→0+ Question 6.3, basically, asks what is the supremum of the topological entropy of geodesic flow (“properly” defined) on singular flat metrics that are conformally equivalent to σ and have total area A. 12 T.BARTHELME´ AND A.ERCHENKO Topological entropy

Metric of constant • negative curvature

Metric entropy

Figure 1. Conjectural possible values of entropies in a ﬁxed conformal class.

While we do not know the answers to the above questions, we expect that the set of < possible pairs (hmetr(g), htop(g)) where g ∈ [σ]A looks like the shaded region on Figure1. By [BE17, Theorem 5.1] and [EK17, Section 2], we see that to increase topological entropy we need to shrink a non-trivial simple closed curve and to preserve negative curvature we need to modify the metric on some neighborhood of that curve whose size, most likely, will depend on the conformal class. Therefore, we do not expect that in a fixed conformal class it is possible to get arbitrary close to the line corresponding to the metric entropy equal 1 −2πχ(M) 2 to A on Figure1. Moreover, the second author and A.Katok showed in [EK17] that any pairs of numbers satisfying (6.1) are realizable as topological and metric entropies of geodesic flow on a set of smooth negatively curved Riemannian metrics with total area A (assumption of g being conformally equivalent to a fixed hyperbolic metric is dropped). In this regard, the geometric constructions in the proof of that fact in [EK17], and having top Corollary 2.5, we expect that lim Hσ (x) exists, and it has to tend to +∞ as σ gets closer x→0+ to the boundary of the Teichmüllerspace. GEOMETRY IN A FIXED CONFORMAL CLASS 13

We would like to point out that we do not expect any additional obstructions in a neigh- 1 −2πχ(M) 2 borhood of the point 0, A because of the following reason. In [Ker80] Kerckhoﬀ

proved that for any Riemannian metrics g1 and g2 the Teichm¨ullerdistance dT eich(g1, g2) between their conformal classes is equal to 1 Eg1 (γ) dT eich(g1, g2) = log sup , (6.2) 2 γ Eg2 (γ)

where γ ranges over all non-trivial simple closed curves (see Definition 2.1 for Eg(γ)). There- fore, C0-closeness of Riemannian metrics implies closeness of their conformal classes in the Teichmüllerspace. In particular, the examples in [EK17, Section 3.1] constructed starting 1 −2πχ(M) 2 from (M, σ) and realizing a neighborhood of the point 0, A belong to conformal classes not far from the conformal class of σ. Therefore, most likely, it is possible to make similar examples in a fixed conformal class.

≤ ≤ 6.2. What is in the compactification of [σ]A? By Theorem 3.2, the set of metrics [σ]A is precompact in the uniform metric sense. Moreover, g is a metric of bounded integral curvature.It seems to be not know how “singular” the metric is. Question 6.4. What are the properties of a metric which is the limit of a sequence of metrics ≤ in [σ]A? 6.3. Flexibility beyond two entropies. There are other interesting and important in- trinsic characteristics of the geodesic flow on negatively curved surfaces beside hmetr(·) and h>(·). Let hharm(g) be the entropy of th geodesic flow on (M, g) with respect to the harmonic invariant measure. Denote by λmax(g) the positive Lyapunov eponent with respect to the measure of maximal entropy. The following inequalities are not between the quantites in a fixed conformal class (see [Rue78], [Kat82], and [Led87]). Let g be a smooth negatively curved Riemannian metric of < non-constant curvature in [σ]A, then 1 −2πχ(M) 2 h (g) < < h (g) < h (g) < λ (g). (6.3) metr A harm top max If any of the inequalities above is replaced by an equality, then all other become equalities and the metric g has constant curvature. < By Corollary 2.5, we know that there exists a uniform upper bound for htop(·) on [σ]A. Therefore, the first natural question is the following.

< Question 6.5. Does there exist a uniform upper bound for λmax(·) on [σ]A? More general question is Question 6.6. What four-tuples of positive numbers satisfying inequalities (6.3) are realiz- < able on [σ]A?

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Department of Mathematics and Statistics, Queen’s University, Kingston, ON E-mail address: [email protected]

Department of Mathematics, Ohio State University, Columbus, OH E-mail address: [email protected]