ON THE WEIL-PETERSSON CURVATURE OF THE MODULI SPACE OF RIEMANN SURFACES OF LARGE GENUS

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1 ON THE WEIL-PETERSSON CURVATURE OF THE MODULI SPACE OF RIEMANN SURFACES OF LARGE GENUS YUNHUI WU Abstract. Let S g be a closed surface of genus g and M g be the moduli space of S g endowed with the Weil-Petersson metric. In this paper we investigate the Weil-Petersson curvatures of M g for large genus g. First, we study the asymptotic behavior of the extremal Weil-Petersson holomorphic sectional curvatures at certain thick surfaces in M g as g. Then we prove two curvature properties on the whole space M g as g in a probabilistic way.. Introduction Let S g be a closed surface of genus g with g >, and M g be the moduli space of S g. Endowed with the Weil-Petersson metric, the moduli space M g is Kähler [Ahl6], incomplete [Chu76, Wol75] and geodesically complete [Wol87]. One can refer to the book [Wol0] for the recent developments on Weil-Petersson geometry. Tromba [Tro86] and Wolpert [Wol86] found a formula for the curvature tensor of the Weil-Petersson metric, which has been applied to study a variety of curvature properties of M g over the past several decades. For examples, the moduli space M g has negative sectional curvature [Tro86, Wol86], strongly negative curvature in the sense of Siu [Sch86], dual Nakano negative curvature [LSY08] and nonpositive definite Riemannian curvature operator [Wu4]. One can also refer to [BF06, Hua05, Hua07a, Hua07b, LSY04, LSY05, LSYY3, Teo09, Wol08, Wol0, Wol, Wolb, WW5] for other aspects of the curvatures of M g. The subject of the asymptotic geometry of M g as g tends to infinity, has recently become quite active: see for examples Mirzakhani [Mir07a, Mir07b, Mir0, Mir3] for the volume of M g, Cavendish-Parlier [CP] for the diameter of M g and Bromberg-Brock [BB4] for the least Weil-Petersson translation length of pseudo-anosov mapping classes. In terms of curvature bounds, by combining the results in Wolpert [Wol86] and Teo [Teo09], we may see that, restricted on the thick part of the moduli space, the scalar curvature is comparable to g as g goes to infinity. The negative scalar curvature can be viewed as the l -norm of the Riemannian Weil-Petersson curvature operator. The l p ( p )-norm of the Weil-Petersson curvature operator was studied in [WW5] as g tends to infinity. For other related topics, one

2 YUNHUI WU can also refer to [FKM3, GPY, LX09, Pen9, RT3, ST0, Zog08] for more details. We focus in this paper on the asymptotic behavior for the Weil-Petersson sectional curvatures as the genus g tends to infinity. Tromba [Tro86] and Wolpert [Wol86] deduced from their formula that the Weil-Petersson holo- π(g ), morphic sectional curvature of M g is bounded above by the constant which confirmed a conjecture of Royden in [Roy75]. If one carefully checks their proofs, this upper bound π(g ) can never be obtained: otherwise, there exists a harmonic Beltrami differential on a closed hyperbolic surface whose magnitude along the surface is a positive constant, which is impossible. As far as we know, the explicit optimal upper bound for the Weil- Petersson holomorphic sectional curvature is not known yet. The aim of this article is to study the Weil-Petersson curvatures for large genus. Our first result tells that the rate g, lying in Tromba-Wolpert s upper bound for Weil-Petersson holomorphic sectional curvature, is optimal as g tends to infinity. More precisely, Theorem.. Given a constant ɛ 0 > ln(3 + ). Let X g M g be a hyperbolic surface satisfying that the injectivity radius inj(x g ) ɛ 0. Then, the Weil-Petersson holomorphic sectional curvature HolK at X g satisfies that max ν HBD(X g) HolK(ν) g where HBD(X g ) is the set of harmonic Beltrami differentials on X g. Buser and Sarnak proved in [BS94] that there exists a universal constant C > 0 such that for all genus g there exists a hyperbolic surface Y g M g such that the injectivity radius inj(y g ) of Y g satisfies that inj(y g ) C ln g. The following corollary is an immediate consequence of Theorem., Buser-Sarnak s above result and Tromba-Wolpert s upper bound for Weil- Petersson holomorphic sectional curvature. Corollary.. The supreme Weil-Petersson holomorphic sectional curvature of the moduli space M g satisfies that sup X g M g max HolK(ν) ν HBD(X g) g. Theorem.8 in [WW5] says that the minimal Weil-Petersson holomorphic sectional curvature of a sufficiently thick hyperbolic surface (sufficiently

3 WEIL-PETERSSON CURVATURE 3 thick means large injectivity radius) is comparable to, which answered a question of M. Mirzakhani. Combine Theorem. with a refinement of the argument for the proof of Theorem.8 in [WW5], we get Theorem.3. Given a constant ɛ 0 > ln(3 + ). Let X g M g be a hyperbolic surface satisfying that the injectivity radius inj(x g ) ɛ 0. Then, the ratio of the minimal Weil-Petersson holomorphic sectional curvature over the maximal Weil-Petersson holomorphic sectional curvature at X g satisfies that min ν HBD(Xg) HolK(ν) max ν HBD(Xg) HolK(ν) g. There are recent suggestions that as the genus g grows large, some regions in the moduli space M g should become increasingly flat. It was shown in [WW5] that this is not true from the view point of Riemannian curvature operator. Actually we showed in [WW5] that the l -norm of the Riemannian Weil-Petersson curvature operator at every point in M g is uniformly bounded below away from zero. It is not known whether this phenomenon still holds for the l -norm of the Riemannian Weil-Petersson sectional curvature. Let X g M g and T Xg M g be the tangent space of M g at X g. For sure T Xg M g is identified with HBD(X g ) which is the set of harmonic Beltrami differentials on X g. Since the rest part of the introduction is on real Riemannian sectional curvatures, with abuse of notation we use T Xg M g instead of HBD(X g ). The following result tells that, from the view point of Riemannian sectional curvature we also have that no region in the moduli space M g becomes increasingly flat as g tends to infinity. The proof of Theorem.4 requires a result due to M. Mirzakhani in [Mir3], which says that a random Riemann surface will contain an arbitrarily large embedded hyperbolic geodesic ball as g tends to infinity. For any two dimensional plane P T Xg M g (maybe not holomorphic), we denote by K(P ) the Riemannian Weil-Petersson sectional curvature of the plane P. Theorem.4. There exists a universal constant C 0 probability satisfies that > 0 such that the lim Prob{X g M g ; min K(P ) C 0 < 0} =. g P T Xg M g The author is grateful to Hugo Parlier for bringing to my attention the Weil-Petersson curvatures on random surfaces.

4 4 YUNHUI WU Since M g has negative sectional curvature [Wol86, Tro86], the following function h is well-defined. h(x g ) := min P T Xg M g K(P ) max P TXg M g K(P ), X g M g. The function h above is also well-defined in any Riemannian manifold of negative (or positive) Riemannian sectional curvature. Recall that Zheng- Yau in [YZ9] proved that a compact Kähler manifold with weakly 4-pinched Riemannian sectional curvature (the range of h is in [, 4)) has nonpositive definite Riemannian curvature operator if the sectional curvature is negative. It is known that the Weil-Petersson metric of M g has negative sectional curvature [Wol86, Tro86] and nonpositive definite Riemannian curvature operator [Wu4]. So it is interesting to study this function h on M g. It is clear that h(x g ) for all X g M g. The results in [Hua05, Wol08] tell that sup Xg M g h(x g ) =. Indeed, one may choose a separating curve α S g and consider the direction along which the length l α pinches to zero. Then the Weil-Petersson holomorphic sectional curvature along the pinching direction will blow up as l α 0 (see [Hua07a, Wol08]). On the other hand, since α is separating, there exists arbitrary flat planes (see [Mas76, Hua05]) near the stratum whose nodes have vanishing α-lengths. Thus, h is unbounded near certain part of the boundary of M g. However, it is not clear about the range of h in the thick part of the moduli space. Our next result is that in a probabilistic way h is unbounded globally on M g as g tends to infinity. More precisely, Theorem.5. For any L > 0, then the probability satisfies that lim Prob{X g M g ; h(x g ) L} =. g Contrast with Zheng-Yau s result in [YZ9], for large enough g, almost no point in the moduli space M g has weakly 4-pinched Riemannian sectional curvature although the Riemannian curvature operator of M g is nonpositive definite [Wu4]. For the proofs of Theorem.,.3,.4 and.5, the main idea is to construct harmonic Beltrami differentials on Riemann surfaces with certain nice properties. The following technique result is crucial in the proofs of all the results above. It is also interesting on itself. Theorem.6. Given a positive integer n Z + and a constant ɛ 0 > ln(3 + ). Let X g M g be a hyperbolic surface. Assume that there exists a set of finite points {p i } n i= X g satisfying that (). inj(p i ) ɛ 0, i n.

5 WEIL-PETERSSON CURVATURE 5 (). dist(p i, p j ) ɛ 0, i j n. Where dist(, ) is the distance function on X g. Then, there exists a harmonic Beltrami differential µ HBD(X g ) such that µ(p i ) µ l (X g), i n. Remark.7. When n = and X g has large injectivity radius, Theorem.6 was obtained in [WW5]. I am kindly told by S. Wolpert that the method in Section of Chapter 8 in his book [Wol0] can also lead to the existence of such a harmonic Beltrami differential for this special case that n = and X g has large injectivity radius. Notation. In this paper, we say f (g) f (g) if there exists a universal constant C > 0, independent of g, such that f (g) C f (g) Cf (g). Plan of the paper. Section provides some necessary background and the basic properties of the Weil-Petersson metric that we will need. In Section 3 we construct the harmonic Beltrami differentials which hold for Theorem.6. We establish Theorem.6 in Section 4 and 5. Then we apply Theorem.6 to prove Theorem. and.3 in Section 6. In Section 7 we will prove Theorem.4 and.5. Acknowledgements are given in the last section.. Notations and Preliminaries In this section we will set our notations and provide some necessary background material on surface theory and Weil-Petersson metric... Hyperbolic disk. Let D be the unit disk in the plane endowed with the hyperbolic metric ρ(z) dz where 4 ρ(z) = ( z ). The distance to the origin is dist D (0, z) = ln + z z. For all r 0, let B(0; r) = {z D; dist D (0, z) < r} and B eu (0; r) = {z D; z < r}. Then, the relation between the hyperbolic geodesic ball and Euclidean geodesic ball is given by the following equation. B(0; r) = B eu (0; er e r + ). Let Aut(D) be the automorphism group of D. For any γ Aut(D) there exist two constants a D and θ [0, π) such that γ(z) = exp(iθ) z a az.

6 6 YUNHUI WU The transitivity of the action of Aut(D) on D tells that for all z D and γ Aut(D), ρ(γ(z)) γ (z) = ρ(z)... Bergman projection. In this subsection we briefly review the formula for the Bergman projection, which is a classical tool to construct harmonic Beltrami differentials on Riemann surfaces. One may refer to [Ahl6] for more details. Let X g be a hyperbolic surface and Γ g be its associated Fuchsian group. A complex-valued function u on D is called a measurable automorphic form of weight 4 with respect to Γ g on D if it is a measurable function on D, and satisfies that u(γ z)γ (z) = u(z), z D, γ Γ g. If we allow a measurable automorphic form u of weight 4 to be holomorphic on D, then we call u is a holomorphic automorphic form of weight 4. We denote by A (D, Γ g ) the complex vector space of all holomorphic automorphic functions of weight 4 with respect to Γ g, which is a (6g 6)- dimensional linear space. Let BL (D, Γ g) be the set of all measurable Beltrami automorphic forms of weight 4 with respect to Γ g on D with f = esssup z D f(z) < where f(z) = u(z) ρ(z) for some measurable automorphic form u(z) of weight 4 with respect to Γ g on D. Recall the Bergman Kernel function K(z, ξ) of the unit disk D is given by K(z, ξ) = π( zξ) = (.) (n + )(n + )(n + 4 3)(zξ)n π n=0 where z and ξ is arbitrary in D. A direct computation gives that (.) K(γ z, γ ξ)γ (z) γ (ξ) = K(z, ξ) for all γ Aut(D). The Bergman projection β of BL (D, Γ g) onto A (D, Γ g ) is given by the following theorem. Theorem. ([Ahl6], Formula (.8)). For any f BL (D, Γ g). Let ξ = x + yi D and set (β f)(z) = f(ξ)k(z, ξ)dxdy, z D. Then we have D β f A (D, Γ g ). Proof. One can also see Theorem 7.3 in [IT9].

7 WEIL-PETERSSON CURVATURE 7.3. Surfaces and Weil-Petersson metric. Let S g be a closed surface of genus g and T g be the Teichmüller space of S g. The tangent space at a point X g = (S g, σ(z) dz ) is identified with the space of harmonic Beltrami differentials on X g which are forms of µ = ψ σ where ψ is a holomorphic quadratic differential on X g. Let da(z) = σ(z)dxdy be the volume form of X g = (S g, σ(z) dz ) where z = x + yi. The Weil-Petersson metric is the Hermitian metric on T g arising from the the Petersson scalar product < ϕ, ψ > W P = S ϕ(z) ψ(z) σ(z) σ(z) da(z) via duality. We will concern ourselves primarily with its Riemannian part g W P. Let Teich(S g ) denote the Teichmüller space endowed with the Weil- Petersson metric. The mapping class group Mod(S g ) acts properly discontinuously on Teich(S g ) by isometries. The moduli space M g of Riemann surfaces, endowed with the Weil-Petersson metric, is defined as M g := Teich(S g )/Mod(S g ). The following proposition has been proved in a lot of literature. For examples one can refer to [Hua07b, Teo09, Wolb]. We use the following form which is proven by Teo through using the Taylor series expansion for a holomorphic function. Proposition. ([Teo09], Proposition 3.). Let X g M g and µ T Xg M g be a harmonic Beltrami differential of X g. Then, for any p X g and 0 < r inj(p), µ(p) C (r) µ(z) da(z) B(p;r) where the constant C (r) = ( 4π 3 ( ( ) 3 ))) and B(p; r) X (+e r ) g is the geodesic ball of radius r centered at p. 4e r Proof. One can also see Proposition.0 in [WW5]. One may refer to [IT9, Wol0] for more details on the Weil-Petersson metric..4. Riemannian tensor of the Weil-Petersson metric. The Weil- Petersson curvature tensor is given by the following. Let µ α, µ β be two elements in the tangent space at X g, and g αβ = µ α µ β da. X g For the inverse of (g ij ), we use the convention g ij g kj = δ ik.

8 8 YUNHUI WU The curvature tensor is given by R ijkl = t k t g l ij gst t k g it t g l sj. Let D = ( ) where is the Beltrami-Laplace operator on X g = (S g, σ(z) dz ). The following curvature formula was established by Tromba and Wolpert independently in [Tro86, Wol86], which has been applied to study various curvature properties of the Weil-Petersson metric in the past thirty years. Theorem.3 (Tromba-Wolpert). The curvature tensor satisfies R ijkl = D(µ i µ j ) (µ k µ l )da + X g D(µ i µ l ) (µ k µ j )da. X g Recall that a holomorphic sectional curvature is a Riemannian sectional curvature along a holomorphic plane. Thus, Theorem.3 gives that Proposition.4 (The formula of holomorphic sectional curvature). Let X g M g and µ T Xg M g. Then the Weil-Petersson holomorphic sectional curvature HolK(µ) along the holomorphic plane spanned by µ is HolK(µ) = X g D( µ ) µ da µ 4. W P We enclose this section by the following proposition, whose proof relies on Proposition., Lemma 5. in [Wola] and the Cauchy-Schwartz inequality. This proposition will be applied several times in this article. The statement is slightly different from Proposition. in [WW5]. Proposition.5. Let X g M g and µ T Xg M g be a harmonic Beltrami differential of X g. Then, the Weil-Petersson holomorphic sectional curvature HolK(µ) satisfies that for any p X g, sup z X µ(z) µ W P HolK(µ) C (inj(p)) µ(p) 4 µ 4 W P where the constant C (inj(p)) > 0 only depends on the injectivity radius inj(p) at p. Proof. It follows from the same argument as the proof of Proposition. in [WW5]. We leave it as an exercise. 3. Construction for the objective harmonic Beltrami differentials In this section we will construct the harmonic Beltrami differentials which hold for Theorem.6.

9 WEIL-PETERSSON CURVATURE 9 First we deal with the case n = in Theorem.6. Let X g M g be a hyperbolic surface, p X g and inj(p) be the injectivity radius of X g at p. For any constant r (0, inj(p)], we consider the characteristic function {, z B(p; r). ν 0 (z) := 0, otherwise. Where B(p; r) X g is the geodesic ball of radius r centered at p. Consider the covering map π : D X g. Up to a conjugation, we lift p to 0 D and let Γ g denote its associated Fuchsian group. Then, it is not hard to see that ν 0 can be lifted to ν 0 HL (D, Γ g) satisfying that for all γ Γ g, (3.) ν 0 (z) := { γ (γ z), γ (γ z) 0, otherwise. z γ B(0; r). We apply the Bergman projection β to ν 0. Lemma 3.. Let ν 0 HL (D, Γ g) given in equation (3.). Then, we have (β ν 0 )(z) = ( er e r + ) γ (z). Proof. The proof is a direct computation. Since 0 < r inj(p), we have γ B(0; r) γ B(0; r) =, γ γ Γ g. Let ξ = x + yi D. Theorem. gives that, for all z D, (β ν 0 )(z) = ν 0 (ξ)k(z, ξ)dxdy D = = Equation (.) tells that γ B(0;r) B(0;r) γ (γ ξ) γ (γ K(z, ξ)dxdy ξ) γ (ξ) γ (ξ) K(z, γ ξ) γ (ξ) dxdy. Recall that K(z, ξ) = K(z, γ ξ) = K(γ z, ξ) γ (γ z)) γ (ξ). n=0 π (n + )(n + )(n + 3)(zξ)n.

10 0 YUNHUI WU Hence, (β ν 0 )(z) = B(0;r) γ (ξ) K(γ z, ξ) (γ (γ z)) γ (ξ) dxdy = ( (n + )(n + )(n + 3) π n=0 (γ (γ z)) (zξ)n dxdy) = π B(0;r) (γ (γ z)) = ( er e r + ) γ (z) where the last equality applies the fact that (γ (γ z)) = B eu(0; er e r + ) dxdy (γ ) (z), γ Γ g. Remark 3.. When the surface has big enough injectivity radius, it was shown in [WW5] that the Weil-Petersson holomorphic sectional curvature along the holomorphic plane spanned by the holomorphic quadratic differential γ (z) is comparable to the maximal Weil-Petersson holomorphic sectional curvature of the moduli space at this surface. Moreover, it is comparable to. Let p, q X g be two points with dist(p, q) r > 0 where r is a constant satisfying that (3.) (3.3) 0 < r min{inj(p), inj(q)}. We lift p and q to 0 and q in D respectively, which satisfies that dist D (0, q) = dist(p, q) r. Let σ q Aut(D) with σ q (0) = q. Actually one may choose σ q (z) = z + q, z D. + qz We define a function ν HL (D, Γ g) as follows. For all γ Γ g, γ (γ z), z γ B(0; r). γ (γ z) γ ν (z) := (γ z) σ q ((γ σ q) z) (3.4), z γ B( q; r). γ (γ z) σ q ((γ σ q) z) 0, otherwise. Equations (3.) and (3.3) tells that ν (z) is well-defined on D.

11 WEIL-PETERSSON CURVATURE Lemma 3.3. For any z D, we have γ (γ ξ) σ q ((γ σ q) ξ) γ (γ ξ) σ q ((γ σ q) ξ) K(z, ξ)dxdy = (σ q γ) (z). γ B( q;r) Proof. Since Γ g Aut(D), γ B( q; r) = γ σ q B(0; r) γ Γ g. Let ξ = x + yi D. Then, for all z D we have γ (γ ξ) γ (γ ξ) σ q ((γ σ q) ξ) σ q ((γ σ q) K(z, ξ)dxdy ξ) = = γ B( q;r) γ σ q B(0;r) B(0;r) = γ (σ q ξ) ( γ Γ B(0;r) γ (σ q ξ) g γ (γ ξ) γ (γ ξ) σ q ((γ σ q) ξ) σ q ((γ σ q) K(z, ξ)dxdy ξ) γ (σ q ξ) σ q (ξ) γ (σ q ξ) σ q (ξ) K(z, γ σ q ξ) (γ σ q ) (ξ) dxdy σ q (ξ) σ q (ξ) K((γ σ q ) z, ξ) ((γ σ q ) ((γ σ q ) z)) ((γ σ q ) (ξ)) (γ σ q) (ξ) dxdy) = ((γ γ Γ σ q ) ((γ σ q ) z)) K((γ σ q ) z, ξ)dxdy g = ( ((γ σ q ) ((γ σ q ) z)) ( (n + )(n + )(n + 3) π n=0 B(0;r) B(0;r) = ((γ γ Γ σ q ) ((γ σ q ) z)) π g ((γ σ q ) z ξ) n dxdy)) = ((γ σ q ) ((γ σ q ) z)) (er e r + ) = ( er e r + ) ((γ σ q ) ) (z) = ( er e r + ) (σ q γ) (z). B eu(0; er e r + ) dxdy

12 YUNHUI WU Now we apply the Bergman projection β to ν (z). First from our assumptions on equations (3.) and (3.3) we know that the balls in {γ B(0; r), γ B( q; r)} γ Γg are pairwisely disjoint. Thus, Lemma 3. and 3.3 tell that Lemma 3.4. For all z D, we have (β ν )(z) = ( er e r + ) ( (σ q γ) (z) + γ (z) ). Similarly we generalize the construction above for any finite subset in X g, which is the remaining part of this section. Given two constants n Z + and ɛ > 0, a finite set of points {p i } n i= X g is called (ɛ, n)-separated if (3.5) dist(p i, p j ) ɛ, i j n. A finite set of points {p i } n i= X g is called an ɛ-net of X g if the set of points {p i } n i= X g are (ɛ, n)-separated and (3.6) n i=b(p i ; ɛ) = X g. Let r > 0 be a constant and {p i } n i= X g be a (r, n)-separated finite set of points satisfying that (3.7) min {inj(p i)} r. i n We lift p to the origin p = 0 D. Let Γ g be its associated Fuchsian group and F be the Dirichlet fundamental domain centered at 0 w.r.t Γ g. We also lift {p i } n i= to {} n i= F respectively. Thus, for all i, j n, (3.8) dist D (, p j ) dist(p i, p j ). For i n, let σ pi Aut(D) with σ pi (0) =. For sure one may choose σ pi (z) = z +, z D. + z In particular σ p is the identity map. That is, σ p (z) = z for all z D. Similar as equation (3.4) we define a function ν n HL (D, Γ g). More precisely, for all γ Γ g and i n, γ (γ z) ν n (z) := σ pi ((γ σ p ) i z), z γ B( p γ (γ z) i; r). (3.9) σ p ((γ σ pi ) z) i 0, otherwise. Proposition 3.5. For any z D, we have (β ν n )(z) = ( er e r + ) (σ γ) (z). i=

13 WEIL-PETERSSON CURVATURE 3 Proof. Since {p i } i n are (r, n)-separated, equation (3.7) tells that γ B( ; r) γ B( p j ; r) =, γ γ Γ g or i j [, n]. Then, the conclusion follows from the same computation as the proof of Lemma 3.3. In the following two sections, we will prove that the harmonic Beltrami differential n i= γ Γg (σ γ) p (z) i dz ρ(z) dz holds for Theorem Two bounds In this section, we use the same notations in Section 3. For each positive integer i [, n], we define (4.) µ i (z) := (σ γ) (z), z D. ρ(z) 4 Where ρ(z) = is the scalar function of the hyperbolic metric on the ( z ) unit disk. The following computation follows from the idea of Ahlfors in [Ahl64] (one can also see [WW5] for an English version). Ahlfors Method: From the triangle inequality we know that (4.) µ i (z) (σ γ) (z). ρ(z) Then since ρ(γ(z)) γ (z) = ρ(z) for any γ Aut(D), and ρ(ζ) = 4( ζ ), we have (4.3) (σ γ) (z) ρ(z) = 4 ( (σ γ)(z) ). The inequalities above yields that for all z D, (4.4) µ i (z) ( (σ 4 γ)(z) ). Let be the (Euclidean) Laplace operator on the (Euclidean) disk. Then a direct computation shows that (4.5) ( ( (σ γ)(z) ) = 8 ( (σ γ(z) ) (σ γ) (z). Note that the terms on the right side are non-negative when σ γ(z). With that in mind, recall that B eu(0; ) := {z D; z < } is the ball of Euclidean radius, let V i := γ Γg γ σ pi B eu (0; ) be the pullbacks of this ball B eu (0; ). The equation above gives that ( (σ

14 4 YUNHUI WU γ)(z) ) is subharmonic in D V i. Since both ( (σ γ)(z) ) and V i are Γ g -invariant, and Γ g is cocompact, we find (4.6) sup z D ( (σ γ)(z) ) = sup γ Γ z V i g ( (σ γ)(z) ) = sup z σ pi B eu(0; ) ( (σ γ)(z) ) which in particular is bounded above by a constant depending on Γ g and. Recall the relation between the Euclidean distance and the hyperbolic distance is dist D (0, z) = ln + z z. Since σ pi Aut(D), σ pi B eu (0; ) is the hyperbolic geodesic ball B( ; ln(3+ )) of radius ln(3 + ) centered at. Hence, equation (4.6) is equivalent to (4.7) sup ( (σ γ)(z) ) = sup z D 4.. A upper bound function. Set z B( ;ln(3+ )) ( (σ γ)(z) ). (4.8) µ(z) = n µ i (z) = i= i= (σ γ) (z) ρ(z), z D. Where {µ i } i n are given in equation (4.). Similar as equation (4.4) we have (4.9) µ(z) 4 i= ( (σ γ)(z) ), z D. Define the right side function to be (4.0) f(z) := ( (σ 4 γ)(z) ), z D. i= From the definition we know that f is a Γ g -invariant function in D, which descends into a function on the hyperbolic surface X g = D/Γ g. Proposition 4.. The function f satisfies that sup f(z) = sup z D z n i= B(;ln(3+ f(z). ))

15 Proof. For i n and z D, set WEIL-PETERSSON CURVATURE 5 f i (z) = ( (σ γ)(z) ). Equation (4.5) tells that the function f i is subharmonic in the complement ( γ Γg γ B( ; ln(3 + ))) c of ( γ Γg γ B(, ln(3 + ))) in D. Since f = n i= f i, we have That is, f(z) 0, z n i=( γ Γg γ B( ; ln(3 + ))) c. f(z) 0, z ( γ Γg n i= γ B( ; ln(3 + ))) c. Since f is Γ g -invariant, it follows from the Maximal-Principal that sup f(z) = sup z D z n i= B(;ln(3+ f(z). )) 4.. Bounds for f when ɛ 0 > ln(3 + ). Given a positive constant ɛ 0 with ɛ 0 > ln(3 + ). Let {p i } i n X g be an (ɛ 0, n)-separated finite set of points satisfying that (4.) min i n inj(p i) ɛ 0. Recall that the origin p = 0 D is a lift of p X g and { } n i= F are the lifts of {p i } n i= respectively, where F is the Dirichlet fundamental domain centered at 0 w.r.t Γ g. In particular, (4.) dist D (, p j ) dist(p i, p j ) ɛ 0, i j n. Lemma 4.. For any z B eu (0; ), there exists a universal positive constant δ, only depending on ɛ 0, such that B eu (z; δ) B(0; ɛ 0 ). Proof. Recall that dist D (0, z) = ln + z z. In particular, we have B eu (0; ) = B(0; ln(3 + )). Since ɛ 0 inequality. > ln(3 + ), the conclusion directly follows from the triangle The following result will be applied to prove Theorem.6.

16 6 YUNHUI WU Proposition 4.3. Given a positive integer n Z + and a constant ɛ 0 > ln(3 + ). Let X g M g be a hyperbolic surface and {p i } i n X g be an (ɛ 0, n)- separated finite set of points satisfying that min inj(p i) ɛ 0 i n. Let µ be the harmonic Beltrami differential given in equation (4.8). Then, (). For any z B eu (0; ) we have µ(z) 6πδ i= Area(σ γ B(0; ɛ 0 )) where δ is the constant in Lemma 4. and Area( ) is the Euclidean area function. (). Evaluated at 0, µ satisfies that µ(0) π (π i= γ e Γ g Area(γ B eu (0; Area(σ γ B eu (0; )) ))). Proof. Proof of Part (). Since σ γ is holomorphic in D for all i n and γ Γ g, ( (σ γ) (z) ) 0, z D. By applying the Mean-Value-Inequality we have, for all z B eu (0; ), f(z) = 4 i= = ( z ) 4 6 = i= 6πδ 6πδ ( (σ γ)(z) ) (σ γ) (z) i= n (σ Area(B eu (z; δ)) γ) (η) dη B eu(z;δ) Area(σ γ B eu (z; δ)) i= Area(σ γ B(0; ɛ 0 )) i= where the last inequality follows from the Lemma 4..

17 WEIL-PETERSSON CURVATURE 7 Then, Part () of the conclusion follows from inequality (4.9) and the inequality above. Proof of Part (). Since σ p is the identity map and ρ(0) = 4, one may rewrite equation (4.8) as (4.3) µ(0) = 4 + γ e Γ g γ (0) + (σ γ) (0). The triangle inequality leads to µ(0) 4 4 ( γ (0) ) n 4 ( (4.4) (σ γ) (0) ). γ e Γ g i= Since (σ γ) (z) is holomorphic in D, we have for all i n and γ Γ g, (4.5) i= (σ γ) (z) 0, z D. By inequality (4.4), (4.5) and the Mean-Value-Inequality, we have µ(0) 4 4 ( (4.6) γ (z) dz ) π/ γ e Γ B g eu(0; ) 4 ( n i= π/ π (π ( ( i= B eu(0; ) γ e Γ g Area(γ B eu (0; Area(σ γ B eu (0; Then, Part () of the conclusion follows. (σ γ) (z) dz ) ))) )))). 5. Proof of Theorem.6 In this section we will prove Theorem.6. Let {p i } i n be the finite set of points in X g satisfying the conditions of Theorem.6. As in last section, we lift p to the origin p = 0 D and also {p i } n i= to {} n i= F respectively, where F is the Dirichlet fundamental domain centered at 0 w.r.t Γ g. Consider µ HBD(X g ) defined in equation (4.8). Then, Theorem.6 is equivalent to the following statement. Theorem 5.. There exists two universal constants C 3, C 4 > 0 such that (). sup z D µ(z) C 3. (). min i n µ( ) C 4.

18 8 YUNHUI WU First we prove Part () of the theorem above. We separate the proof into several lemmas. The first one is elementary in hyperbolic geometry. Recall that Area( ) is the Euclidean area function. Lemma 5.. Let B(0; r) be the hyperbolic geodesic ball of radius r centered at 0 where r > 0. Then, for any h Aut(D) we have Area(h B(0; r)) = Area(h B(0; r)). Proof. Since h Aut(D), there exists θ [0, π) and a D such that h(z) = exp (iθ) z a, z D. az Then, we have h a + exp ( iθ)z (z) =, z D. + a exp ( iθ)z Use the area transformation formula we have (5.) Area(h B(0; r)) = h (z) dz (5.) Similarly we have Area(h B(0; r)) = B(0;r) B(0;r) = ( a ) B(0;r) = ( a ) B(0;r) (h ) (η) dη B(0;r) az 4 dz. + a exp ( iθ)η 4 dη. After taking a substitution z = exp ( iθ)η in B(0; r), it is clear that (5.3) az 4 dz = + a exp ( iθ)η 4 dη. Then, the conclusion follows from equations (5.), (5.) and (5.3). Lemma 5.3. For either γ γ Γ g or i j [, n], γ σ pi B(0; ɛ 0 ) γ σ pj B(0; ɛ 0 ) =. Proof. Since σ pi Aut(D), we have σ pi B(0; ɛ 0 ) = B(; ɛ 0 ). Case (a). i j [, n]. For any γ, γ Γ g, we project the geodesic balls {γ B( ; ɛ 0 ), γ B( p j ; ɛ 0 )} D to the two balls {B(p i ; ɛ 0 ), B(p j ; ɛ 0 )} in X g = D/Γ g. Since we assume that dist(p i, p j ) ɛ 0, B(p i ; ɛ 0 ) B(p j; ɛ 0 ) =

19 which in particular implies WEIL-PETERSSON CURVATURE 9 γ B( ; ɛ 0 ) γ B( p j ; ɛ 0 ) =. Case (b). γ γ Γ g and i = j. For this case the geodesic balls {γ B( ; ɛ 0 ), γ B( ; ɛ 0 )} in D are the two lifts of the geodesic ball B(p i ; ɛ 0 ) X g. Then, the conclusion follows from our assumption that inj(p i ) ɛ 0. Proof of Part () of Theorem 5.. First inequality (4.9) and Proposition 4. tell that (5.4) sup µ(z) sup z D z n i= B(;ln(3+ f(z) )) where f is given in equation (4.0). (5.5) Recall that p = 0. First we show that sup z B(0;ln(3+ f(z) )) 6δ where δ is the universal constant in Lemma 4.. For any z B(0; ln(3 + )), let δ be the universal constant in Lemma 4.. Then, (5.6) B eu (z; δ) B(0; ɛ 0 ). Combine Part () of Proposition 4.3 and Lemma 5., we have for all z B(0; ln(3 + )), n (5.7) f(z) 6πδ Area(γ σ pi B(0; ɛ 0 )). i= Lemma 5.3 tells that the balls {γ σ pi B(0; ɛ 0 )} i n,γ Γg are pairwisely disjoint. Hence, inequality (5.7) tells that for all z B(0; ln(3 + )), (5.8) f(z) 6πδ Area(γ σ pi B(0; ɛ 0 )) i= 6πδ Area(D) = 6δ. (5.9) Since z is arbitrary in B(0; ln(3 + )), we have sup z B(0;ln(3+ f(z) )) 6δ.

20 0 YUNHUI WU We continue to prove Part () of Theorem 5.. For any i 0 [, n] and z B(0 ; ln(3 + )). So we have z = σ pi0 (η) for some η B(0; ln(3 + )). Since ρ(σ pi0 (η)) σ 0 (η) = ρ(η), we have (5.0) f(z) = f(σ pi0 (η)) n i= γ Γ = g (σ γ) (σ pi0 (η)) ρ(σ pi0 (η)) n i= γ Γ = g (σ γ σ pi0 ) (η)). ρ(η) Since η B(0; ln(3 + )), by using the same argument in the proof of Part () of Proposition 4.3 we have f(z) 6πδ i= Area((σ γ σ pi0 ) B(0; ɛ 0 )) From Lemma 5. we have n (5.) f(z) 6πδ Area((σ 0 γ σ pi ) B(0; ɛ 0 )) i= Lemma 5.3 tells that the balls {γ σ pi B(0; ɛ 0 )} i n,γ Γg are pairwisely disjoint. Since σ 0 Aut(D), the geodesic balls {σ 0 γ σ pi B(0; ɛ 0 )} i n,γ Γg are also pairwisely disjoint. Hence, inequality (5.) tells that for all z B(0 ; ln(3 + )), (5.) f(z) = 6πδ i= 6πδ Area(D) 6δ. Since i 0 [, n] is arbitrary, we have (5.3) Area((σ 0 γ σ pi ) B(0; ɛ 0 )) sup z n i= B(;ln(3+ f(z) )) 6δ. Then, Part () of the conclusion follows from inequalities (5.4), (5.9) and (5.3) by choosing C 3 =. 6δ

21 WEIL-PETERSSON CURVATURE Proof of Part () of Theorem 5.. Recall p = 0. We first show that (5.4) µ(0) ((eɛ0 e ɛ 0 + ) ). Recall that ɛ 0 > ln(3 + ). So the constant satisfies that ɛ 0 ((e e ɛ 0 + ) ) > 0. Recall the Euclidean ball B eu (0; ) is the same as the hyperbolic disk B(0; ln(3 + )). Then, Lemma 5. tells that for all i n and γ Γ g, (5.5) Area(σ γ B eu (0; )) = Area(γ σ pi B eu (0; )). From Lemma 5.3 and equation (5.5) we know that (5.6) + Area(γ B eu (0; )) γ e Γ g Area(γ σ pi B eu (0; )) i= γ Γ g Area(D) Area(B(0; ɛ 0 )) = π( ( e ɛ0 e ɛ 0 + ) ). Thus, from Part () of Proposition 4.3 and inequality (5.6) we know that (5.7) µ(0) π (π ɛ 0 π( (e e ɛ 0 + ) )) = ((eɛ0 e ɛ 0 + ) ). We continue to prove Part () of Theorem 5.. For any i 0 [, n] and we let σ pi Aut(D) with σ pi0 (0) = 0. Then, (5.8) µ(0 ) = µ σ pi0 (0).

22 YUNHUI WU Since ρ(σ pi0 (0)) σ 0 (0) = ρ(0) = 4, from equation (4.8) and the triangle inequality we know that (5.9) µ(0 ) = i= (σ 4 4 ( γ) (σ pi0 (0)) ρ(σ pi0 (0)) (σ 0 γ σ pi0 ) (0) ) γ e Γ g 4 ( (σ γ σ pi0 ) (0) ). i i 0 Similar as the proof of Part () of Proposition 4.3 we have (5.0) µ(0 ) 4 4 π ( γ e Γ g 4 π ( i i 0 = 4 π ( π ( i i 0 By Lemma 5. we have (5.) µ(0 ) 4 π ( π ( i i 0 B eu(0; ) B eu(0; ) Since B eu (0; ) B(0; ɛ 0 ), we have (5.) (σ 0 γ σ pi0 ) (z) dz ) (σ γ σ pi0 ) (z) dz ) Area(σ 0 γ σ pi0 B eu (0; γ e Γ g Area(σ γ σ pi0 B eu (0; ))) ))). Area(σ 0 γ σ pi0 B eu (0; γ e Γ g Area(σ 0 γ σ pi B eu (0; ))) ))). µ(0 ) 4 π ( Area(σ 0 γ σ pi0 B(0; ɛ 0 ))) γ e Γ g π ( Area(σ 0 γ σ pi B(0; ɛ 0 ))). i i 0 Since σ pi0 Aut(D), from Lemma 5.3 we know that for all either γ γ Γ g or i j [, n] we have (5.3) σ 0 γ σ pi B(0; ɛ 0 ) σ 0 γ σ pj B(0; ɛ 0 ) =.

23 WEIL-PETERSSON CURVATURE 3 Thus, equations (5.) and (5.3) lead to (5.4) µ(0 ) 4 (Area(D) Area(σ π 0 e σ pi0 B(0; ɛ 0 ))) = 4 ɛ 0 (π π(e π e ɛ 0 + ) ) = ((eɛ0 e ɛ 0 + ) ). Since i 0 [, n] is arbitrary, Part () of the conclusion follows from inequalities (5.4) and (5.4) by choosing C 4 = ((eɛ0 e ɛ 0 + ) ). 6. Proof of Theorem. and.3 In this section we will use the harmonic Beltrami differential µ defined in equation (4.8) to prove Theorem. and Theorem.3. Proposition 6.. Given a positive integer n Z + and a constant ɛ 0 > ln(3 + ). Let X g M g be a hyperbolic surface and {p i } i n X g be an (ɛ 0, n)- separated finite set of points satisfying that min inj(p i) ɛ 0 i n. Let µ be a harmonic Beltrami differential given in equation (4.8). Then, µ W P n. Proof. We lift p to the origin p = 0 D. Let Γ g be its associated Fuchsian group, F be a Dirichlet fundamental domain centered at 0 w.r.t Γ g and { } i n F be the lifts of {p i } i n respectively. Since {p i } i n X g = D/Γ g be an (ɛ 0, n)-separated and ɛ 0 ln(3 + ), the triangle inequality tells that (6.)B( ; ln(3 + )) B( p j ; ln(3 + )) =, i j [, n]. Since min i n inj(p i ) ɛ 0, we have (6.) B( ; ln(3 + )) F, i [, n]. First we prove the upper bound.

24 4 YUNHUI WU Equations (6.) and (6.) tell that (6.3) µ W P = µ(z) ρ(z) dz F n i= B( ;ln(3+ )) µ(z) ρ(z) dz. Since inj(p i ) ɛ 0 > ln(3 + ), from Proposition. and Part () of Theorem 5. we know that n µ (6.4) W P C (ln(3 + )) µ() i= n i= = n C (ln(3 + )) C 4 C 4 C (ln(3 + )). Now we prove the lower bound. From Part () of Theorem 5. we know that (6.5) µ W P = µ(z) ρ(z) dz F µ l (D) µ(z) ρ(z) dz F C 3 µ(z) ρ(z) dz. (6.6) Inequality (4.) tells that µ W P C 3 F F i= (σ γ) (z) dz = C 3 Area(σ γ F ) i= n = C 3 Area(D) i= = n (C 3 π). Then, the conclusion follows from inequalities (6.4) and (6.6). Now we are ready to prove Theorem.. Proof of Theorem.. Let {p i } i n be an ɛ 0 -net in X g where n is a positive integer to be determined.

25 (6.7) (6.8) (6.9) WEIL-PETERSSON CURVATURE 5 First since dist(p i, p j ) ɛ 0 for all i j [, n], we have Thus, B(p i ; ɛ 0 ) B(p j; ɛ 0 ) =, i j [, n]. n i= Since inj(x g ) ɛ 0, Thus, we have Vol(B(p i ; ɛ 0 )) = Vol( n i=b(p i ; ɛ 0 )) Vol(X g ) = 4π(g ). Vol(B(p i ; ɛ 0 )) = Vol D(B(0; ɛ 0 )). n 4π(g ) Vol D (B(0; ɛ 0 )). On the other hand, since {p i } i n X g is an ɛ 0 -net, n i=b(p i ; ɛ 0 ) = X g. Since inj(x g ) ɛ 0, after taking a volume we get Thus, 4π(g ) = Vol(X g ) n Vol(B(p i ; ɛ 0 )) n i= = Vol D (B(0; ɛ 0 )) n. 4π(g ) Vol D (B(0; ɛ 0 )). Inequalities (6.7) and (6.8) tell that n g. We choose µ HBD(X g ) defined in equation (4.8). Recall that Proposition.5 says that the Weil-Petersson holomorphic sectional curvature along µ satisfies that (6.0) HolK(µ) sup z X g µ(z) µ. W P Proposition 6. and equation (6.9) tell that (6.) µ W P g.

26 6 YUNHUI WU Then, it follows from Theorem 5., inequality (6.0) and equation (6.) that the Weil-Petersson holomorphic sectional curvature along µ satisfies that (6.) HolK(µ) C 5 g where C 5 > 0 is a universal positive constant. In particular, we have (6.3) max HolK(ν) C 5 ν HBD(X g) g. On the other hand, from Wolpert-Tromba s upper bound for Weil-Petersson holomorphic sectional curvature in [Wol86, Tro86] we know that (6.4) max HolK(ν) ν HBD(X g) π(g ). Then, the conclusion follows from inequalities (6.3) and (6.4). The following result is a refinement of Theorem.8 in [WW5]. Theorem 6.. Given a positive constant ɛ > ln(3 + ). Let X g M g be a hyperbolic surface satisfying that there exists a point p X g such that inj(p) ɛ. Then, there exists a universal constant C 6 = C 6 (ɛ ) > 0, only depending on ɛ, such that the minimal Weil-Petersson holomorphic sectional curvature at X g satisfies that min HolK(ν) C 6 < 0. ν HBD(X g) Proof. We lift p X g to the origin 0 D. Let Γ g be its associated Fuchsian group and µ HBD(X g ) given by µ(z) = γ (z) ρ(z) which agrees with equation (4.8) for the case n =. Recall that Proposition.5 says that there exists a constant C > 0 such that (6.5) HolK(µ) C (inj(p)) µ(p) 4 µ 4. W P Since inj(p) ɛ > ln(3 + ), by applying Part () of Theorem 5. and Proposition 6. to µ for the case n =, we have (6.6) µ(p) µ W P. Then, the conclusion immediately follows from inequality (6.5) and equation (6.6). Now we are ready to prove Theorem.3.

27 WEIL-PETERSSON CURVATURE 7 Proof of Theorem.3. Since inj(x g ) ɛ 0 > ln(3 + ), by Theorem. in [Hua07b] (or Theorem. in [WW5]) and Theorem 6. we know that (6.7) min HolK(ν). ν HBD(X g) Then, the conclusion follows from Theorem. and equation (6.7). 7. Proof of Theorem.4 and.5 Before proving Theorem.4 and.5, let us recall the following two results of M. Mirzakhani in [Mir3] which are crucial in this section. Given a constant ɛ > 0, let M ɛ g = {X g M g ; inj(x g ) ɛ}. Theorem 7. ([Mir3], Theorem 4.). There exists a universal constant D 0 > 0 such that for all ɛ < D 0, as g. Let X be a hyperbolic surface. Set Vol W P (M ɛ g) ɛ Vol W P (M g ) Emb(X) = max p X inj(p). Theorem 7. ([Mir3], Theorem 4.5). lim Prob{X g M g ; Emb(X g ) ln g g 6 } =. Proof of Theorem.4. It is clear that the conclusion directly follows from Theorem 6. and Theorem 7.. Proof of Theorem.5. Let C 6 > 0 be the universal constant in Theorem.4. Define A g := {X g M g ; min K(P ) C 6 }. P T Xg M g First a result of Teo in [Teo09] (see Proposition 3.3 in [Teo09]) tells that for any X g (M g M ɛ g) and v T Xg M g, the Ricci curvature Ric(v) along the v direction satisfies that (7.) Ric(v) C (ɛ) where the constant C is given in Proposition.. Since Ric is a (6g 7) summation, inequality (7.) tells that (7.) (7.3) That is, (6g 7) max P T Xg M g K(P ) C (ɛ). max K(P ) C (ɛ) P T Xg M g 6g 7.

28 8 YUNHUI WU Thus, it follow from inequality (7.3) and the definition of A g that for any X g A g (M g M ɛ g), (7.4) h(x g ) C 6 (6g 7). C (ɛ) Let D 0 > 0 be the constant in Theorem 7.. Inequality (7.4) tells that for any L > 0 and any 0 < ɛ D 0 there exists a positive integer g 0 >> such that for all g g 0 we have (7.5) h(x g ) L, X g A g (M g M ɛ g). Meanwhile, the Weil-Petersson volume of A g (M g M ɛ g) is controlled as follows. Vol W P (A g (M g M ɛ g)) (7.6) Vol W P (M g ) = Vol W P (A g ) + Vol W P (M g M ɛ g) Vol W P (A g (M g M ɛ g)). Vol W P (M g ) Vol W P (A g ) Vol W P (M g ) Vol W P (M ɛ g) Vol W P (M g ). Thus, it follows from Theorem.4, inequality (7.6) and Theorem 7. that there exists a universal constant C 7 > 0 such that (7.7) (7.8) lim inf g Vol W P (A g (M g M ɛ g)) C 7 ɛ. Vol W P (M g ) Combine inequalities (7.5) and (7.7), we get lim sup Prob{X g M g ; h(x g ) L} g lim inf g M g ; h(x g ) L} g lim inf g C 7 ɛ. Vol W P (A g (M g M ɛ g)) Vol W P (M g ) Then, the conclusion follows because ɛ (0, D 0 ) is arbitrary,. 8. Acknowledgement This paper is an outgrowth of work done in collaboration with Michael Wolf who I would like to especially thank. Without the invaluable discussions with him, it is impossible to have this work done. The author also would like to thanks Zheng Huang, Maryam Mirzakhani and Scott Wolpert for their interests and useful conversations. This work was partially completed while the author attended the Tsinghua Sanya Group Action Forum on Dec/04. The author would like to thank the organizers for their hospitality.

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