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On Asymptotic Weil-Petersson Geometry of Teichm\"{u}ller Space of Riemann Surfaces

In this paper, we study the asymptotic geometry of Teichmuller space of Riemann surfaces and give bounds on the Weil-Petersson sectional curvature of Teichmuller space, in terms of the length of the shortest geodesic on the surface. This will also imply that the sectional curvature is not pinched from above or below by any negative constant.

on a surface. The moduli problem of Riemann asks how many distinct complex structures can exist on a closed surface, and this problem has been developed into the modern theory of Teichmüller space.
In this paper, we assume Σ is a smooth, closed Riemann surface of genus g, with n punctures and 3g − 3 + n > 1. Teichmüller space T g,n is the space of hyperbolic metrics (with constant curvature −1) on Σ, where two hyperbolic metrics σ and ρ are equivalent if there is a biholomorphic map between (Σ, σ) and (Σ, ρ) in the homotopy class of the identity map.
Teichmüller space T g,n has its own natural complex structure ( [1]): T g,n is a complex manifold of complex dimension 3g − 3 + n > 1, the cotangent space at Σ is identified with QD(Σ), the space of holomorphic quadratic differentials; and the tangent space at Σ is identified with the space of so-called harmonic Beltrami differentials.
The Weil-Petersson metric on Teichmüller space is naturally defined by duality from the L 2 inner product on QD(Σ). This metric is considered as one of the natural metrics on Teichmüller space. Every Weil-Petersson isometry of Teichmüller space is induced by an element of the extended mapping class group when 3g − 3 + n > 1 and (g, n) = (1, 2) ( [14]). And the Weil-Petersson metric has many interesting geometric properties: for example, it is a Riemannian metric with negative sectional curvature ( [25] [21]); yet it is incomplete, since not every geodesic can be extended indefinitely, the surface Σ developes a node when a geodesic cannot be further extended ( [13] [5] [24]); this metric is Kähler ( [1]), and there is a negative upper bound −1 2π(g− 1) , which only depends on the topology of the surface, for the holomorphic sectional curvature and Ricci curvature ( [25]); however, there are no negative upper bounds for the sectional curvature ( [10]).
Our goal in this paper is to investigate the asymptotic geometry of the Weil-Petersson metric on Teichmüller space and give an estimate on the upper and lower bounds for the Weil-Petersson sectional curvature at any point in Teichmüller space, purely in terms of the length of the shortest geodesic on the surface: Theorem 1.1. Let l be the length of the shortest geodesic on closed surface Σ, and K be the Weil-Petersson sectional curvature of Teichmüller space T , assuming dim C T > 1, there exists a constant C > 0 such that Moreover, there are tangent planes with the Weil-Petersson curvatures of the orders O(l) and comparable to l −1 , and hence the Weil-Petersson sectional curvature has neither negative upper bound, nor lower bound.
We will firstly prove a result on asymptotic holomorphic sectional curvature. Our estimate indicates that the holomorphic sectional curvature tends to negative infinity when a core geodesic on the surface is performed infinitesimal twists and length shrinking. More specifically, we show that Theorem 1.2. If the complex dimension of Teichmüller space T is greater than 1, then there is no negative lower bound for the holomorphic sectional curvature of the Weil-Petersson metric. Moreover, let l be the length of the shortest geodesic along a path to the frontier of Teichmüller space, then there exists a sequence of tangent planes with Weil-Petersson holomorphic sectional curvature of the order comparable to l −1 .
We realized that one of the difficulties in estimating curvatures is working with the operator D = −2(∆ − 2) −1 , which appears in Tromba-Wolpert's curvature formula. As pointed out in ( [22]), there is a natural correspondence of the operator D and local variations of the energy of a harmonic map between surfaces. By investigating harmonic maps from a nearly noded surface to nearby hyperbolic structures, in [10], we showed that even though the sectional curvatures are negative, they are not staying away from zero. More specifically, we detected an asymptotically flat tangent plane, denoted by Ω l , spanned by two Beltrami differentialṡ µ 0 andμ 1 , resulting from pinching two independent core geodesics on the surface. Theorem 1 of [10]. If the complex dimension of Teichmüller space T is greater than 1, then the Weil-Petersson sectional curvature is not pinched from above by any negative constant. Moreover, the Weil-Petersson sectional curvature of Ω l is of the order O(l).
In theorem 1.2, we are detecting a tangent plane, spanned by Beltrami differentialsμ 0 and iμ 0 , whose curvature is asymptotically negative infinity. Following a suggestion of Scott Wolpert, we consider a family of tangent planes Ω ′ l , spanned by iμ 0 andμ 1 , and find that Theorem 1. 3. Ω ′ l is asymptotically flat, i.e., its Weil-Petersson curvature is of the order O(l).
In other words, we are detecting another asymptotically flat tangent plane, spanned by iμ 0 andμ 1 . This asymptotic flatness results from pinching two nonhomotopic closed geodesics on the surface while performing infinitesimal twists on one of them. Together with theorem 1 of ( [10]), this suggests an asymptotic product structure of Weil-Petersson metric, as pointed out by Wolpert ([26]). Similarly, we also show that: Theorem 1.4. The plane Ω ′′ l spanned by Beltrami differentials iμ 0 and iμ 1 is asymptotically flat with respect to the Weil-Petersson metric, and its Weil-Petersson sectional curvature is of the order O(l).
We notice that theorems 1.3, 1.4 and theorem 1 of ( [10]) are theorems concerning asymptotic flatness, and the path we take towards the frontier of Teichmüller space (see definition of the frontier of Teichmüller space in §2.1) is to pinch two short independent geodesics on the surface. We can also pinch just one simple closed curve on the surface to take a path towards the frontier space. There is also a phenomenon of asymptotic flatness when a separating geodesic on the surface is pinched. When Σ is a closed surface with genus at least two and γ 0 is a separating short geodesic on Σ with length l. Let γ 2 and γ 3 be two closed geodesics on Σ which have fixed length l 0 >> l, and these two geodesics lie on different sides of the shrinking curve γ 0 . We can define a tangent plane Ω ′′′ l , spanned by Beltrami differentialsμ 2 andμ 3 , whereμ 2 andμ 3 are obtained from infinitesimal twists about curves γ 2 and γ 3 , respectively. Theorem 1.5. This plane Ω ′′′ l is asymptotically flat with respect to the Weil-Petersson metric, moreover, its Weil-Petersson sectional curvature is of the order O(l).
Here is the more detailed content of this paper. We give the necessary background in section 2. Section 3 is devoted to proving theorem 1.2. The discussion of this purpose is broken into subsections: in §3.1, we study a harmonic mapping problem between hyperbolic cylinders. The family of rotationally symmetric harmonic maps will be an approximation to the actual family of harmonic maps restricted on the pinching neighborhood. We will describe so called "model case one" in §3.2, namely, we pinch one core geodesic of a cylinder into a point, and perform infinitesimal twists about this geodesic, and study the asymptotic behavior of the harmonic maps between cylinders. We will establish the estimates of terms in the curvature formula in model case one; in §3.3, we construct families of maps which have small tension, and are close to the harmonic maps resulting from pinching and twisting process in §3.2; finally in §3.4, we prove theorem 1.2 based on the estimates in §3.2 and the construction in §3. 3. In section 4 we study the asymptotic flatness of the Weil-Petersson metric with two nonhomotopic geodesics on the surface are shrinking. We describe model case two in §4.1, estimate curvature terms in §4.2, and prove theorems 1.3 and 1.4 in §4.3. The aim of section 5 is to prove curvature bounds for the Weil-Petersson metric. In §5.1, we consider the asymptotic flatness when only one separating geodesic on the surface is pinched, and prove theorem 1.5; We prove theorem 1.1 in §5.2.
Acknowledgements The author expresses his deepest thanks to Mike Wolf for his mentorship, continuous encouragement and many fruitful discussion. Theorem 1.1 was brought up to the author by Scott Wolpert, the author also wants to thank him for many helpful discussion. The author also thanks Yair Minsky for suggesting him to prove theorem 1.5.

Harmonic Maps and Teichmüller Space
We will give some background in this section.
By the uniformization theorem, the set of all similarly oriented hyperbolic structures M −1 can be identified with the set of all conformal (or complex) structures on Σ with the given orientation. And Teichmüller space T is defined to be the quotient space The moduli space of Riemann surfaces admits the Deligne-Mumford compactification ( [16]), and any element of the compactification divisor can be thought of as a Riemann surface with nodes, a connected complex space where points have neighborhoods complex isomorphic to either {|z| < ε} (regular points) or {zw = 0; |z|, |w| < ε} (nodes). We can think of noded surfaces arising as elements of the compactification divisor through a pinching process: fix a family of simple closed curves on the surface Σ such that each component of the complement of the curves has negative Euler characteristic. Topologically, the noded surface is the result of identifying each curve to the node ( [3]).
Teichmüller space T is a complex manifold when 3g − 3 + n > 1, and the cotangent space at a point Σ ∈ T is the space of holomorphic quadratic differentials Φdz 2 on Σ ( [1]). The Weil-Petersson metric on T is defined on QD(Σ) ∼ = T * Σ T by the L 2 -norm: σ dzdz where σ|dz| 2 is the hyperbolic metric on Σ. By duality, we obtain a Riemannian metric on the tangent space to T .
Teichmüller space with Weil-Petersson metric is not complete ([5] [24]). The incompleteness is caused by pinching of at least one short geodesic on the surface. For the surface with genus at least two, we defineT to be the Weil-Petersson completion of T , and denote ∂T as the frontier setT \T . Hence as shown in ( [13]), every point in ∂T represents a noded surface, i.e., the frontier set ∂T consists of a union of lower dimensional Teichmüller spaces, each such space consists of topologically reduced Riemann surfaces (noded surfaces), obtained by pinching nontrivial geodesics on the surface.
The curvature tensor of the Weil-Petersson metric is given by ( [25]) where dA is the area element andμ's are infinitesimal Beltrami differentials. Here the operator D = −2(∆ − 2) −1 . It is known that the operator D is a self-adjoint compact integral operator with a positive kernel, and it is the identity on constant functions. Then the curvature of Ω is then given by R/Π, where ( [25]) It is known that the Weil-Petersson sectional curvature, holomorphic sectional curvature and Ricci curvature are all negative ([25] [21]).

Harmonic Maps and Local Variations
Our main method is to look at families of harmonic maps between degenerating Riemann surfaces. The method of harmonic maps has been intensively studied as an important computational tool in understanding the geometry of Teichmüller space. In particular, the second variation of the energy of the harmonic map w = w(σ, ρ) with respect to the domain structure σ (or image structure ρ) at σ = ρ yields the Weil-Petersson metric on T ( [21], [22]), and one can also re-establish Tromba-Wolpert's curvature tensor formula of the Weil-Petersson metric from this method ( [11], [22]).
A harmonic map is a critical point of the energy fuctional E(w; σ, ρ); it satisfies the Euler-Lagrange equation, namely, w zz + ρw ρ w z wz = 0. The Euler-Lagrange equation for the energy is the condition for the vanishing of the tension, which is, in local coordinates, [18]) that given σ, ρ, there exists a unique harmonic map w : (Σ, σ) → (Σ, ρ) homotopic to the identity of Σ, and this map is in fact a diffeomorphism. Naturally associated to a harmonic map w : (Σ, σ|dz| 2 ) → (Σ, ρ|dw| 2 ) is a quadratic differential Φ(σ, ρ)dz 2 , which is holomorphic with respect to the conformal structure of σ. This association of a quadratic differential to a conformal structure then defines a map Φ : T → QD(Σ) from Teichmüller space T to the space of holomorphic quadratic differentials QD(Σ). This map Φ is a homeomorphism ( [22]).
Thus we have a holomorphic quadratic differential Φdz 2 = ρw zwz dz 2 , and evidently where σ = ρ means that (Σ, σ) and (Σ, ρ) are the same point in Teichmüller space T . Also note that the map w can be extended to surfaces with finitely many punctures.
We define two auxiliary functions as following: When we restrict ourselves to the situation when K(σ) = K(ρ) = −1, we will have the following facts ( [22]): • The energy density is e = H + L; • The Jacobian is H − L; where H = 0. Now we consider a family of harmonic maps w(t) for t small, where w(0) = id, the identity map. Denote by Φ(t) the family of Hopf differentials determined by w(t). We rewrite ∆logH = 2H − 2L − 2 as The maximum principle will force all the odd order t-derivatives of the holomorphic energy H(t) to vanish, since the above equation only depends on the modulus of Φ(t) and not on its argument( [22]), and H(t) is real-analytic in t ( [23]). For the family of harmonic maps w(t), the domain hyperbolic structure is fixed and the target metric is changing. Wolf computed the t-derivative of various geometric quantities associated with this family w(t), and we collect these local variational formulas into: •Ḣ(t) = ∂/∂t α | 0 H(t) = 0; With this lemma, and recall that D = −2(∆ − 2) −1 , we obtain a partial differential equation aboutḦ(t) which will play an important role in the proof of our theorems:

Holomorphic Sectional Curvature
In this section, we will focus on estimating the Weil-Petersson holomorphic sectional curvature, and prove theorem 1.2, namely, If the complex dimension of Teichmüller space T is greater than 1, then there is no negative lower bound for the holomorphic sectional curvature of the Weil-Petersson metric. Moreover, let l be the length of the shortest geodesic along a path to a boundary point in Teichmüller space, then there exists a sequence of tangent planes with Weil-Petersson holomorphic sectional curvature of the order comparable to l −1 .
We organize this section into following subsections. In §3.1, we will study harmonic maps between hyperbolic cylinders, i.e., we consider a family of pertubations of the identity map between two cylinders; in §3.2, we estimate terms in the curvature formula in the model case one where the surface is a long cylinder; in §3.3, we construct a family of C 2,α maps between surfaces and show that the constructed maps have small tension and are close to the harmonic maps we obtain from the pinching and twisting process; in §3.4, we adapt the estimates in §3.2 and §3.3 to prove theorem 1.2.
For the sake of simplicity of exposition, we assume that our surface has no punctures. We will discuss the case when finitely many punctures are allowed in §3.5.

Harmonic Maps between Cylinders
In this subsection, we consider the asymptotics of harmonic maps, as pertubations of the identity map, between two hyperbolic cylinders. In particular, using the same notation in [10], we study the boundary value problem of harmonically mapping the cylinder where the hyperbolic length element on M is lcsc(lx)|dz|, to the cylinder with boundary identification where the hyperbolic length element on N is Lcsc(Lu)|dw|. Here l and L are the lengths of the simple closed core geodesics in the corresponding cylinders.
Let w = u + iv be this harmonic map between cylinders M and N , where The Euler-Lagrange equation becomes Note that both M and N admit an anti-isometric reflection about the curves is independent of x, and c 0 (l, l) = 0 since u(x) is the identity map when L = l.
So we have So the solution to the Euler-Lagrange equation can be derived from the equation It is not hard to show that when l → 0, the solution u(l, L; x) converges to a solution u(L; x) to the "noded" problem, i.e., M = [1, +∞), where we require u(L; 1) = L −1 sin −1 (L) and lim x→+∞ u(L; x) = π 2L . This "noded" problem has the explicit solution as following ( [23])

Model Case One
Consider the surface Σ which is developing a node, i.e., we are pinching one short closed geodesic γ 0 on Σ to a point p 0 . We denote M 0 its pinching neighborhood, i.e., a cylinder described as M in §3.1 centered at γ 0 . In our model case one, the surface is this cylinder M 0 with core geodesic γ 0 .
We define M (l, θ) be the surface with two of the Fenchel-Nielsen coordinates, namely, the hyperbolic length of γ 0 being l and the twisting angle of γ 0 being θ. We obtain a point M (l) = M (l, 0) in Teichmüller space T g . Note that as l tends to zero, the surface is developing a node. Fix M (l), we vary the length of γ 0 into length L = L(t), where L(0) = l. Thus we obtain a family of harmonic maps W 0 (t) : M (l) → M (L(t), 0) = M (L(t)). The t-derivative of µ 0 (t), the Beltrami differential of the the family W 0 (t), at t = 0 represents a tangent vector,μ 0 , of Teichmüller space T g at M (l); And a tangent vector iμ 0 is obtained by performing infinitesimal twists θ(t) on the shrinking curve γ 0 , where Hence these two tangent vectorṡ µ 0 and iμ 0 will span a two dimensional subspace of the tangent space T M (l) T g to T g , therefore we obtain a family, Ω * l , of tangent planes.
Proposition 3.1. The Weil-Petersson holomorphic sectional curvatures of Ω * l is of the order comparable to l −1 . Hence the Weil-Petersson curvatures tend to negative infinity as l tends to zero.
It is easy to see that Proposition 3.1 implies our theorem 1.2.
We denote φ 0 (t) as the Hopf differential corresponding to the cylinder map w 0 (t) in M 0 . Here w 0 (t) : M 0 (l) → M 0 (L(t)) is a family of harmonic maps described as w = u(x) + iy in §3.1. In other regions of the surface, we abuse our notations a little bit and still use φ 0 to denote the Hopf differential corresponding to harmonic map W 0 (t). We also denote µ 0 as the corresponding Beltrami differential to φ 0 .
We denote a = a(l) = l −1 sin −1 (l), is never zero for all positive l, otherwise, we would haveẇ z = 0 asφ = σẇ z and hence w is a constant map by rotational invariance of the map.
Note that most of the mass of |φ 0 | resides in the thin part associated to γ 0 , near where the core geodesic γ 0 is pinched, and that is why the estimate in the long cylinder is critical in our calculation.
We will now estimate all terms in the curvature formula in this model case one where we consider the surface as a long cylinder M 0 . In M 0 , the corresponding Beltrami differential iṡ . The Weil-Petersson holomorphic sectional curvature is given by the quotient −R 0000 /|μ 0 | 4 . And we will estimate 1/|μ 0 | 4 and |R 0000 | in the cylinder M 0 in the next two lemmas.
We recall in M 0 , the Hopf differential φ 0 is corresponding to the cylinder map w 0 (t) : (M 0 , σ) → (M 0 , ρ(t)), and |φ 0 | = 1. As in §2.2 and §3.1, we write the holomorphic energy H = ρ(w(z)) σ(z) |w z | 2 , therefore we can write D(|μ 0 | 2 ) =Ḧ. Then in the cylinder M 0 , we have A maximum principle argument implies thatḦ is positive. ThisḦ converges to the holomorphic energyḦ 0 in the "noded" problem (of §3.1) when x is fixed but sufficiently large in [a, π/2l]. This convergence guarantees thatḦ is bounded on the compacta in [a, b] and so we can with the conditions Recall from §3.1 that all the odd order t-derivatives of H(t) vanish. Also notice that is a particular solution to equation (2). Hence we can check, by the method of reduction of the solutions, the general solution to equation (2) with the assigned conditons has the form where coefficients A 2 = A 2 (l) and A 3 = A 3 (l) are constants independent of x and we can check, by substituting the solution into the assigned conditions, that A 2 and A 3 are of the order comparable to l −1 .
We compute the following This completes the proof of Lemma 3.3.

Construction of Maps
In last subsection, we estimated the terms in the curvature formula in the model case one where the surface is a long cylinder. Essentially, in the pinching neighborhood M 0 , we used the cylinder map w 0 (rotationally symmetric harmonic map) instead of the actual harmonic map W 0 during the computation. Now we are in the general setting, i.e., the surface Σ is developing a node. In this subsection, we will construct a family of maps G 0 (t) to approximate the harmonic map W 0 (t), and the essential parts of this family are the identity map of the surface restricted in the non-cylindral part and the cylinder map in the pinching neighborhood. We will also show that this constructed family G 0 (t) is reasonably close to the harmonic maps W 0 (t); hence we can use the estimates we obtained in the previous subsection to the general situation.
We recall some of the notations from previous subsections. We still set M 0 to be the pinching neighborhood of the node p 0 . Also W 0 (t) is the harmonic map corresponding to pinching γ 0 in M 0 into length L = L(t), where L(0) = l, and twisting γ 0 with angle θ(t), where θ(0) = 0. Let w 0 (t) be cylinder maps in model case one, we want to show that W 0 (t) is close to w 0 (t) in M 0 and is close to identity map outside of M 0 .
An important feature of twisting the core geodesic γ 0 is that it will change the values of the correspondingḦ on the boundary of the long cylinder M 0 . However, we recall from last subsection that the convergence ofḦ toḦ 0 in the noded problem guaranteesḦ is bounded on the boundary of M 0 . Therefore we can still use model case one to include the consideration of twisting the core geodesic, since we do not concern the boundary values ofḦ, as long as they are bounded. And in the compact region, where is far away from the shrinking and twisting curve γ 0 , and changes of valueḦ will be small, as we will see in next subsection.
We note that G 0 consists of 3 parts. It is the cylinder map of M 0 deep into the cylinder region in M 0 ∩Σ 0 , the identity map in the compact region Σ 0 \M 0 , and a smooth map in the intersection region B(∂M 0 , 1). Among three parts of the constructed map G 0 (t), two of them, the identity map and the cylinder map, are harmonic hence have zero tension; thus the tension of G 0 (t) is concentrated in B(∂M 0 , 1). From §3.2, for the cylinder map w 0 = u(x) + iy, we have Thus we can require that |g t,z | ≤ C 2 t and |g t,zz | ≤ C 2 t, and the constant C 2 = C 2 (t, l) is bounded in both t and l since the coefficient of t for c 0 (t) is bounded for small t and small l. With the local formula of the tension in §2.2, we have τ (G 0 (t)), the tension of G 0 (t) is of the order O(t). Note that these constructed maps G 0 (t) are not necessarily harmonic.
Now we are about to compare the constructed family G 0 (t) and the family of harmonic maps W 0 (t). To do this, we consider the following function Q 0 = cosh(dist(W 0 , G 0 )) − 1.
is bounded for small t and l.
Proof. First, we claim that Q 0 is a C 2 function. Notice that both the harmonic map W 0 (t) and the constructed map G 0 (t) are the identity map when t = 0, and both families vary smoothly without changing homotopy type in t for sufficiently small |t| ( [6]). For all l > 0, and for any ε > 0, there exists a δ such that for |t| < δ, we have |W 0 (t) − W 0 (0)| < ε 2 and |G 0 (t)−G 0 (0)| < ε 2 . Therefore the triangular inequality implies that |W 0 (t) − G 0 (t)| < ε. Since l is positive, the Collar Theorem ( [4]) implies that the surface has positive injectivity radius r bounded below, and we choose our ε << r, then Q 0 is well defined and smooth.
We follow an argument in [8]. For any unit v ∈ T 1 (B (∂M 0 , 1)), the map G 0 satifies the inequality , then we find that for any x ∈ Σ, where γ x is the geodesic joining G 0 (x) to W 0 (x) with initial tangent vec- If G 0 (t) does not coincide with W 0 (t) on B(∂M 0 , 1), we must have all maxima of Q 0 (t) on the interior of B(∂M 0 , 1), at any such maximum, we apply the inequality |dG 0 (v)| 2 > 1 − ε 0 to (12) to find so that at a maximum of Q 0 , we have We notice that Q 0 is of the order dist 2 (W 0 , G 0 ) and sinhdist(W 0 , G 0 ) is of the order dist(W 0 , G 0 ), this implies that dist(W 0 , G 0 ) is of the order O(t) in B(∂M 0 , 1), which completes the proof of Lemma 3.4.

Proof of Theorem 1.2
In last subsection, we constructed a family of maps and found that these maps are reasonably close to the actual harmonic maps between surfaces. Now we are ready to adapt the estimates in the model case one to the general setting, and prove the Proposition 3.1, which will imply theorem 1.2.
Proof of Proposition 3.1. We notice that Lemma 3.2 still holds from triangle inequality and Lemma 3.6. It will be sufficient to show that Lemma 3.3 still holds, which implies desired curvature estimate immediately.
Now we are about to estimate Σ D(|μ 0 | 2 )|μ 0 | 2 σdxdy, which breaks into 2 integrals as follows: where K is the compact set disjoint from M 0 ∩ Σ 0 . For the last integral, from the previous discussion, because of the convergence of the harmonic maps to the harmonic maps of "noded" problem, we have both |μ 0 | and |μ 1 | are bounded. The maximum principle implies that D(|μ 0 | 2 ) =Ḧ ≤ sup{|μ 0 | 2 }. Note that K is compact, hence we have the second integral is of the order of O(1).
Recall that (∆ − 2)Ḧ G = −2 where H G is the holomorphic energy corresponding to the model case one when the harmonic map is the cylinder map, and φ G 0 is the quadratic differential corresponding to the constructed map G 0 . We also denote µ G 0 to be the Beltrami differential corresponding to φ G 0 . Also recall, from Lemma 3.6, that is bounded in l for small l. So we can set some λ = O(1) (bounded in l for small l) such that |φ 0 | 2 < λ 2 |φ G 0 | 2 and at the boundary of M 0 ∩ Σ 0 satisfiesḦ < λḦ G . For example, we can take λ  lxcot(lx)). Now we find that, Recalling the computation in §3.3, we have the following: These add up to Σ D(|μ 0 | 2 )|μ 0 | 2 σdxdy = O(l −7 ), with Lemma 3.2, we have the holomorphic sectional curvature is of the order O(l 6 )O(l −7 ) = O(l −1 ).

Surface with Punctures
In last four subsections, we proved theorem 1.2. We want to point out here that the assumption on the surface Σ being compact is not essential. In other words, our theorem 1.2, as well as other theorems in this paper, are still true when the surface Σ has finitely many punctures. Now we prove theorem 1.2 in the case of Σ being a punctured surface.
The existence of a harmonic diffeomorphism between punctured surfaces has been investigated by Wolf ([23]) and Lohkamp ([12]). In particular, Lohkamp ([12]) showed that a homeomorphism between punctured surfaces is homotopic to a unique harmonic diffeomorphism with finite energy, and the holomorphic quadratic differential corresponding to the harmonic map in the homotopy class of the identity is a bijection between Teichmüller space of punctured surfaces and the space of holomorphic quadratic differentials.
In this case, the set Σ\(M 0 ∩ Σ 0 ) is no longer compact. Let K 0 be a compact surface with finitely many punctures, and {K m } be a compact exhaustion of K 0 . We now estimate K 0 D(|μ 0 | 2 )|μ 0 | 2 dA. Let H(t) be the holomorphic energy corresponding to the harmonic map w(t) : (K 0 , σ) → (K 0 , ρ(t)), then H(t) is bounded from above and below, and has nodal limit 1 near the punctures ( [23]), hence |μ 0 | 2 has the order o(1) near the punctures. To see this, we consider K 0 as the union of K m and disjoint union of finitely many punctured disks, each equipped hyperbolic metric |dz| 2 z 2 log 2 z . Then |μ 0 | = O(|zlog 2 z|) → 0 as z tends to the p uncture, since the quadratic differential has a pole of at most the first order. We notice that ∂K 0 is the boundary of the cylinders, where the harmonic maps converge to a solution to the "noded" problem as l → 0, hence D(|μ 0 | 2 ) =Ḧ(t) is bounded on ∂K 0 . Therefore we apply Omori-Yau maximum principle ( [17] [28]) to the differential equation (∆ − 2)Ḧ = −2|μ 0 | 2 on K 0 and obtain that In other words, our proof carries over to the punctured case, which completes the proof of theorem 1.2.
We point out that similar argument can apply to the proof of other theorems of this paper when finitely many punctures are allowed. For this reason, we will assume in the rest of the paper that our surface has no punctures.

Asymptotic Flatness I: Two Curves Pinching
In this and next sections, we aim to prove theorem 1.1. Firstly we want to discuss the asymptotic flatness of the Weil-Petersson metric on Teichmüller space, when there are at most two curves on the surface are shrinking. We deal with the two curves pinching case in this section.
As indicated in §2.1, the frontier space ∂T of Teichmüller space is a union of lower dimensional Teichmüller spaces, each consists of noded surfaces obtained by pinching nontrival geodesics on the surface. Hence to obtain two tangent vectors near the infinity, at least one geodesic on the surface is being pinched. In §4.1, we describe model case two where the surface is a pair of cylinders; in §4.2, we prove theorem 1.3 in model case two; we give general proofs of theorems 1.3 and 1.4 in §4.3.

Model Case Two
In this subsection, following [10], we discuss the phenomena of asymptotic flatness of the Weil-Petersson resulting from pinching two independent curves on the surface.
When we pinch two nonhomotopic curves γ 0 and γ 1 on surface Σ to two points, say p 0 and p 1 , the surface Σ is developing two nodes. We denote M 0 and M 1 pinching neighborhoods for these two geodesics, i.e., two cylinders described as M in §3.1, centered at γ 0 and γ 1 , respectively.
We define M (l 0 , l 1 ) be the surface with two of the Fenchel-Nielsen coordinates, namely, the hyperbolic lengths of γ 0 and γ 1 , are l 0 and l 1 , respectively. When we set the length of these two geodesics equal to l simultaneously, we will have a point M (l) = M (l, l) in Teichmüller space T g . As l tends to zero, the surface is developing two nodes. At this point M (l), there are two tangent vectors iμ 0 andμ 1 . Hereμ 0 is the same as described in §3.2, i.e., we fix γ 1 in M 1 having length l, and pinch γ 0 in M 0 into length L = L(t), where L(0) = l. So the t-derivative of µ 0 (t) at t = 0 represents a tangent vector,μ 0 , of Teichmüller space T g at M (l), and iμ 0 is the tangent vector obtained by performing infinitesimal twists on the curve γ 0 , as seen in §3.2. We denote the resulting harmonic map by W 0 (t) : M (l, l) → M (L(t), l). Similarly when we fix γ 0 in M 0 having length l, and pinch γ 1 into length L = L(t), the t-derivative of µ 1 (t) at t = 0 represents another tangent vector,μ 1 , at M (l); we denote the resulting harmonic map by W 1 (t) : M (l, l) → M (l; L(t)). We obtain a family of tangent planes Ω ′ l , spanned by iμ 0 andμ 1 . Now we estimate the curvatures of Ω ′ l , the result is: Theorem 1.3. Ω ′ l is asymptotically flat, i.e., its Weil-Petersson sectional curvature is of the order O(l).
Again, as in §3.2, we will still use rotational symmetric harmonic maps to approximate harmonic maps in the cylinder regions. We describe our "model case two": the surface is a pair of hyperbolic cylinders M 0 and M 1 , as we are pinching two independent curves γ 0 and γ 1 .
We denote φ 0 (t) as the Hopf differential corresponding to the cylinder map w 0 (t) in M 0 , and φ 1 (t) as the Hopf differential corresponding to w 1 (t) in M 1 . Here w 0 (t) : M 0 (l) → M 0 (L(t)) and w 1 (t) : M 1 (l) → M 1 (L(t)) are harmonic maps described as w = u(x)+iy in §3.1. We still use φ 0 to denote the Hopf differential corresponding to harmonic map W 0 (t) in M 1 , and φ 1 as the Hopf differential corresponding to harmonic map W 1 (t) in M 0 .
Therefore the corresponding Beltrami differentials are:
From Lemmas 4.1 and 4.2, we will need to establish next estimate to prove theorem 1.3: Proof. The surface is a pair of two cylinders, so we need to estimates two integrals: For the first integral M 0 D(|iμ 0 | 2 )|μ 1 | 2 dA, we write D(|μ 0 | 2 ) = H, where H(t) is the holomorphic energy for the family of harmonic maps w 0 (t), as in §3.2. Then in M 0 , we have (∆ − 2)Ḧ = −2 |φ 0 | 2 σ 2 = −2l −4 sin 4 (lx). We recall thatḦ is bounded on the compacta in [a, b] and so we can assume that A 1 (l) =Ḧ(a) =Ḧ(l −1 sin −1 l) = O(1) > 0. ThenḦ(x) solves the differential equation with the conditions We remark here that infinitesimal twists only change the boundary values forḦ, which will stay bounded.
We have solved above differential equation in §3.2, and the general solutions have the following form where coefficients A 2 = A 2 (l) and A 3 = A 3 (l) are constants that satisfy with the conditions Ḧ (l −1 sin −1 l) = B 1 (l),Ḧ(π/l − l −1 sin −1 l) = B 2 (l).
Here B 1 (l) and B 2 (l) are positive and bounded as l tends to zero, sincë H converges to the holomorphic energy in the "noded" problem (of §3.2, Consider the equation with the boundary conditions that satisfy We choose h(x) = −x −2 and claim thatḦ − h is a subsolution to (9) for x ∈ [a, b]. To see this, noticing that 2|φ 0 | 2 | M 1 l −4 sin 4 (lx) decays rapidly as x → b for small l, we have Notice that λY (x) solves the equation (9) if Y (x) does, for any constant λ. So up to multiplying by a bounded constant, we have Y | ∂M 1 > (Ḧ − h(x))| ∂M 1 . HenceḦ − h is a subsolution to (9) for x ∈ [a, b], while the solutions to (9) have the form where constants B 3 = B 3 (l) and B 4 = B 4 (l) satisfy, from the boundary conditions for the equation (9), that Now we compute the second integral: This proves Lemma 4.3.
Combined with the estimates of Lemmas 4.1, 4.2 and 4.3, we proved theorem 1.3 when the surface is a pair of cylinders.

Proof of Theorems 1.3 and 1.4
We are now ready to complete the proof of theorem 1.3. Recall from §3.3, we constructed a family of maps G 0 (t) to approximate W 0 (t). The map G 0 (t) consists of three parts: the cylinder map w 0 of M 0 deep into the cylinder region in M 0 ∩ Σ 0 , the identity map in the compact region Σ 0 \M 0 , and a smooth map interplaying with the identity map and w 0 in the intersection region B(∂M 0 , 1). Similarly, we obtain a family of maps G 1 (t) such that it is w 1 in M 1 ∩ Σ 1 , the identity map in Σ 1 \M 1 , and a smooth map in B(∂M 1 , 1). Here Σ 0 = {p ∈ Σ : dist(p, ∂M 0 ) > 1}, and B(∂M 0 , 1) = {p ∈ Σ : dist(p, ∂M 0 ) ≤ 1}.
Proof of Theorem 1.3. Parallel to the discussion in §3.3, we know that both families G 0 (t) and G 1 (t) are close to the families of harmonic maps W 0 (t) and W 1 (t), respectively (Lemma 3.4). Also It is not hard to see that Lemma 4.1 still holds, so we need to establish the same estimate as in Lemma 4.3 in order to prove theorem 1.3 in the general case, i.e., we need to estimate Σ D(|iμ 0 | 2 )|μ 1 | 2 σdxdy, which breaks into three integrals as following: is the solution to (∆ − 2)Y = 0, and constants B 3 and B 4 satisfy that B 3 = O(l) and B 4 = O(l). Apply this to (12): From the calculation in §3.3, and we have the following: Applying these estimates to (12), we have . This and Lemma 4.1 complete the proof of theorem 1.3.

Asymptotic Flatness II: Curvature Bounds
The goal of this section is to prove theorem 1.5 and, finally, theorem 1.1. We continue to discuss the asymptotic flatness of the Weil-Petersson metric when there is only one shrinking geodesic on the surface in §5.1, then give a proof of theorem 1.1 in §5.2.
In this section, we always use l to denote the length of the shortest geodesic on the surface.

One Curve Pinching
In this subsection, we assume that Σ is a closed surface of genus g at least two, and γ 0 is a closed separating short geodesic on the surface Σ with length l. As l tends to zero, the surface is developing a separating node. Let γ 2 and γ 3 be two closed geodesics on the different sides of curve γ 0 . Without loss of generality, we assume l(γ 2 ) = l(γ 3 ) >> l.
We denote M 0 as a cylinder centered at γ 0 , while M 2 and M 3 are cylinders centered at γ 2 and γ 3 , respectively. Note that as the length of γ 0 is kept very short, the cylinder M 0 becomes very long and similar to the cylinder described as M in §3.1. Now we have two families of harmonic maps: the family W 2 (t) : M (l) → M (l, θ 2 (t), 0), which fixes γ 3 , keeps γ 0 very short at length l, and twists γ 2 at angle θ 2 (t); and the family W 3 (t) : M (l) → M (l, θ 3 (t), 0), which fixes γ 2 , keeps γ 0 very short at length l, and twists γ 3 at angle θ 3 (t). We denoteμ 2 andμ 3 as infinitesimal Beltrami differentials corresponding to W 2 (t) and W 3 (t), respectively. Alsoφ 2 andφ 3 are corresponding infinitesimal Hopf differentials. Now we obtain a tangent plane Ω ′′′ l at a point M (l) = M (l, 0, 0) in Teichmüller space, spanned by tangent vectorsμ 2 andμ 3 , and we will show that Theorem 1.5. This plane Ω ′′′ l is asymptotically flat with respect to the Weil-Petersson metric, moreover, its Weil-Petersson sectional curvature is of the order O(l).
To show |R| = O(l), we notice that |R| ≤ C ′ Σ D(|μ 2 | 2 )|μ 3 | 2 σdxdy for some positive constant C ′ , as in the proof of Lemma 3 of [10] and Lemma 4.2. Hence we need to calculate the integral Σ D(|μ 2 | 2 )|μ 3 | 2 dA. Since γ 0 is a separating curve, we split this integral into two parts, one for each side of γ 0 , and we abuse our notation to denote the side with curve γ 2 by M 2 , and the other side with curve γ 3 by M 3 .
Since M 2 is compact, and |φ 3 | M 2 = O(l 4 ), we have the integral Also M 3 is compact, we have |φ The estimate of Lemma 5.1, (13) and (14) complete the proof of theorem 1.5 when we consider the family of harmonic maps being rotationally symmetric. Now we consider the families of harmonic maps which are no longer rotationally symmetric. Since γ 0 separates the surface, the cylinder M 0 separates the surface into two sides. We denote the side containing γ 2 by M 2 , and the side containing γ 3 by M 3 . As we indicated, the infinitesimal Hopf differentials are exponentially small in the thin part. Thus we can still assume that |φ 2 | M 3 = O(l 4 ) and |φ 3 | M 2 = O(l 4 ), hence the term Π in the curvature formula is still bounded.
Remark 5.2. If the shrinking curve γ 0 (l) is not separating, the twisting neighborhoods M 2 and M 3 will lie on the same component in the limit of surfaces with one shrinking curve as l tends to zero. It is not hard to see that K(Ω ′′′ l ) is now bounded away from zero.
Remark 5.3. Theorem 1.5 can be easily generalized to treat the case when multiple closed geodesics are pinching and at least one of them is separating. Infinitesimal twists about curves on different sides of a separating and shrinking curve will give a family of asymptotically flat tangent planes.
If there is a lower bound for the length of the shortest closed geodesics on Σ, the Weil-Petersson sectional curvature of Teichmüller space is bounded since all integrals in curvature terms are bounded away from negative infinity and zero. Hence, to consider large absolute value of the sectional curvature, we can assume one of the vectorsν 0 andν 1 is corresponding to a deformation of the length of a short geodesic on the surface. Recalling that the Hopf differential not corresonding to pinching this curve will be exponentially small in the thin part of this shrinking curve, so above integral Σ D(|ν 0 | 2 )|ν 1 | 2 dA will be no more than the integral Σ D(|ν 0 | 2 )|iν 0 | 2 dA, which we estimated in the proof of theorem 1.2. Therefore the proof of theorem 1.2 implies the following theorem, which is the lower bound part of theorem 1.1, i.e., Theorem 5.4. Let l be the length of the shortest closed geodesic on the surface, then there is a positive constant C such that the Weil-Petersson sectional curvature K of Teichmüller space satisfies K ≥ −(Cl) −1 , if the complex dimension of Teichmüller space is great than one. a = a(l) = l −1 sin −1 (l), and b = b(l) = πl −1 − l −1 sin −1 (l). Therefore we find that D(|ν 0 | 2 ) is of the order of O(lcot(lb)) = O(l) in Σ\M 0 . Hence, Note that Σ\M 0 |ν 0 | 2 dA = O(1), we have the absolute value of the curvature is given by We notice that the integral M 0 D(|ν 0 | 2 )|ν 1 | 2 dA is positive, and will be smaller when the infinitesimal Beltrami differentialsν 1 is resulting from a deformation of the length of a core geodesic, independent of γ 0 , on the surface, than otherwise. To see this, we recall that |φ 1 |, whereφ 1 is the infinitesimal Hopf differential corresponding toν 1 , decays exponentially away from the curve where the deformation occurs. Therefore |φ 1 | is smaller in M 0 when it is actually resulting from a deformation of the length of a core geodesic γ 1 than the case of otherwise. Now the proofs of theorem 1 of This completes the proof of theorem 5.6. Theorem 1.1 immediately follows from theorem 5.4 and theorem 5.6.