Jeffrey F. Brock
Given a closed hyperbolic 3-manifold T_ψthat fibers over the circle with monodromy ψ: S -> S, the monodromy $ψ$ determines an isometry of Teichmuller space with its Weil-Petersson metric whose translation distance ||ψ||_WP is positive. We show there is a constant K >= 1 depending only on the topology of S so that the volume of T_ψsatisfies ||ψ||_WP/K <= vol(T_ψ) <= K ||ψ||_WP.
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In this paper we study the injectivity radius based at a fixed point along Weil-Petersson geodesics. We show that the square root of the injectivity radius based at a fixed point is $ 0.3884$-Lipschitz on Teichmüller space endowed with the Weil-Petersson metric. As an application we reprove that the square root of the systole function is uniformly Lipschitz on Teichmüller space endowed with the Weil-Petersson metric, where the Lipschitz constant can be chosen to be $0.5492$. Applications to asymptotic geometry of moduli space of Riemann surfaces for large genus will also be discussed.
A brief history of the investigation of the Weil-Petersson curvature and a summary of Teichmüller theory are provided. A report is presented on the program to describe an intrinsic geometry with the Weil-Petersson metric and geodesic-length functions. Formulas for the metric, covariant derivative and formulas for the curvature tensor are presented. A discussion of methods is included. Recent and new applications are sketched, including results from the work of Liu-Sun-Yau, an examination of the Yamada model metric and a description of Jacobi fields along geodesics to the boundary.
We show that the strong asymptotic class of Weil-Petersson (WP) geodesics with narrow end invariant and bounded annular coefficients is determined by the forward ending lamination. This generalizes the Recurrent Ending Lamination Theorem of Brock-Masur-Minsky. As an application we provide a symbolic condition for divergence of WP geodesic rays in the moduli space.
We extend a theorem of Masur and Wolf which says that given a hyperbolic surface S, every isometry of the Teichmuller space for S with the Weil-Petersson metric is induced by an element of the mapping class group for S. Our argument handles the previously untreated cases of the four-holed sphere, the one-holed torus, and the two-holed torus.
We prove that every Weil-Petersson isometry of the Teichmuller space T(g,n) is induced by an element of the extended mapping class group; here 3g-3+n > 1 and (g,n) is not (1,2). Our method follows Ivanov's proof of the Royden's analogous theorem for the Teichmuller metric: we study the action of an isometry on the frontier of the metric completion of the Teichmuller space, and show that the isometry then induces an automorphism on the relevant complex of curves. Some synthetic geometry of the Weil-Petersson metric completes the proof.
Scalar curvature, volume, and Weil-Petersson translation length
本文在伪阿诺索夫映射环面上构造满足特定条件的度量，改进了相关不等式的常数，给出了伪阿诺索夫映射环面双曲体积与Weil-Petersson平移长度关系的新证明。
Making use of the dual Bonahon-Schläfli formula, we prove that the dual volume of the convex core of a quasi-Fuchsian manifold $M$ is bounded by an explicit constant, depending only on the topology of $M$, times the Weil-Petersson distance between the hyperbolic structures on the upper and lower boundary components of the convex core of $M$.
Uniform bounds are developed for derivatives of solutions of the $2$-dimensional constant negative curvature equation and the Weil-Petersson metric for the Teichmüller and moduli spaces. The dependence of the bounds on the geometry of the underlying Riemann surface is studied. The comparisons between the $C^0$, $C^{2,α}$ and $L^2$ norms for harmonic Beltrami differentials are analyzed. Uniform bounds are given for the covariant derivatives of the Weil-Petersson curvature tensor in terms of the systoles of the underlying Riemann surfaces and the projections of the differentiation directions ont…
In this article we show that for every finite area hyperbolic surface $X$ of type $(g,n)$ and any harmonic Beltrami differential $μ$ on $X$, then the magnitude of $μ$ at any point of small injectivity radius is uniform bounded from above by the ratio of the Weil-Petersson norm of $μ$ over the square root of the systole of $X$ up to a uniform positive constant multiplication. We apply the uniform bound above to show that the Weil-Petersson Ricci curvature, restricted at any hyperbolic surface of short systole in the moduli space, is uniformly bounded from below by the negative reciprocal of the…
In general, it is difficult to measure distances in the Weil-Petersson metric on Teichmüller space. Here we consider the distance between strata in the Weil-Petersson completion of Teichmüller space of a surface of finite type. Wolpert showed that for strata whose closures do not intersect, there is a definite separation independent of the topology of the surface. We prove that the optimal value for this minimal separation is a constant $δ_{1,1}$ and show that it is realized exactly by strata whose nodes intersect once. We also give a nearly sharp estimate for $δ_{1,1}$ and give a lower bound …
A 2008 general overview on Weil-Petersson geometry is offered. A preliminary plan for the subsequent CBMS lectures at Central Connecticut State University is included. Mirzakhani's solution of Witten-Kontsevich is not included - this work essentially requires its own lectures. Lectures on Mirzakhani's Witten-Kontsevich were given at the 2011 Park City Mathematics Institute. Research on WP geometry coming after 2008 includes the Brock-Masur-Minsky second paper, Asymptotics of WP geodesics II: bounded geometry and unbounded entropy; the Burns-Masur-Wilkinson, The WP geodesic flow is ergodic; and…
The complex hyperbolic form as a Weil-Petersson form
本文在穿孔球面模空间上建立了复双曲形式（Kähler形式）与Weil-Petersson形式之间的新等式，通过将模空间元素解释为球面上的奇异欧几里得度量来实现。
We prove that any two finite-area non-compact hyperbolic Riemann surfaces S and T have finite covers that are arbitrarily close in the normalized Weil-Petersson metric, where we normalize by dividing the square of the metric by the area of the surface. In the case where T is the modular surface this reduces to showing that S has a finite cover with a proper ideal triangulation where most of the shear coordinates are small; we will construct such a cover out of a random collection of immersed ideal triangles in S.
In this paper we study the systole function along Weil-Petersson geodesics. We show that the square root of the systole function is uniformly Lipschitz on Teichmüller space endowed with the Weil-Petersson metric. As an application, we study the growth of the Weil-Petersson inradius of moduli space of Riemann surfaces of genus $g$ with $n$ punctures as a function of $g$ and $n$. We show that the Weil-Petersson inradius is comparable to $\sqrt{\ln{g}}$ with respect to $g$, and is comparable to $1$ with respect to $n$. Moreover, we also study the asymptotic behavior, as $g$ goes to infinity, of t…
We use ending laminations for Weil-Petersson geodesics to establish that bounded geometry is equivalent to bounded combinatorics for Weil-Petersson geodesic segments, rays, and lines. Further, a more general notion of non-annular bounded combinatorics, which allows arbitrarily large Dehn-twisting, corresponds to an equivalent condition for Weil-Petersson geodesics. As an application, we show the Weil-Petersson geodesic flow has compact invariant subsets with arbitrarily large topological entropy.
Given a Riemann surface with boundary S, the lengths of a maximal system of disjoint simple geodesic arcs on S that start and end at the boundary of S perpendicularly are coordinates on the Teichmueller space T(S). We compute the Weil-Petersson Poisson structure on T(S) in this system of coordinates and we prove that it limits pointwise to the piecewise-linear Poisson structure defined by Kontsevich on the arc complex of S. As a byproduct of the proof, we obtain a formula for the first-order variation of the distance between two closed geodesics under Fenchel-Nielsen deformation.
In this paper we prove that the limit set of any Weil-Petersson geodesic ray with uniquely ergodic ending lamination is a single point in the Thurston compactification of Teichmüller space. On the other hand, we construct examples of Weil-Petersson geodesics with minimal nonuniquely ergodic ending laminations and limit set a circle in the Thurston compactification.
Let $S_g$ be a closed surface of genus $g$ and $\mathbb{M}_g$ be the moduli space of $S_g$ endowed with the Weil-Petersson metric. In this paper we investigate the Weil-Petersson curvatures of $\mathbb{M}_g$ for large genus $g$. First, we study the asymptotic behavior of the extremal Weil-Petersson holomorphic sectional curvatures at certain thick surfaces in $\mathbb{M}_g$ as $g \to \infty$. Then we prove two curvature properties on the whole space $\mathbb{M}_g$ as $g\to \infty$ in a probabilistic way.
The Weil-Petersson diameter of moduli space for large genus
该研究确定固定穿孔数下，模空间的Weil-Petersson直径随亏格的渐近阶为$\boldsymbol{\text{sqrt}}(g)\boldsymbol{\text{log}}(g)$，核心贡献是提出带厚端点且内部无systole界的极小测地线的积分Ricci估计。
An expansion is developed for the Weil-Petersson Riemann curvature tensor in the thin region of the Teichmüller and moduli spaces. The tensor is evaluated on the gradients of geodesic-lengths for disjoint geodesics. A precise lower bound for sectional curvature in terms of the surface systole is presented. The curvature tensor expansion is applied to establish continuity properties at the frontier strata of the augmented Teichmüller space. The curvature tensor has the asymptotic product structure already observed for the metric and covariant derivative. The product structure is combined with t…
In this paper we study the Weil-Petersson geometry of $\overline{\mathcal{M}_{g,n}}$, the compactified moduli space of Riemann surfaces with genus g and n marked points. The main goal of this paper is to understand the growth of the diameter of $\overline{\mathcal{M}_{g,n}}$ as a function of $g$ and $n$. We show that this diameter grows as $\sqrt{n}$ in $n$, and is bounded above by $C \sqrt{g}\log g$ in $g$ for some constant $C$. We also give a lower bound on the growth in $g$ of the diameter of $\overline{\mathcal{M}_{g,n}}$ in terms of an auxiliary function that measures the extent to which …
We introduce a method for constructing Weil-Petersson (WP) geodesics with certain behavior in the Teichmüller space. This allows us to study the itinerary of geodesics among the strata of the WP completion and its relation to subsurface projection coefficients of their end invariants. As an application we demonstrate the disparity between short curves in the universal curve over a WP geodesic and those of the associated hyperbolic $3$-manifold.
On a convex body in a Euclidean space, we introduce a new variational formulation for its Funk metric, a Finsler metric compatible with the tautological Finsler structure of the convex body. We generalize the metric on Teichmuller spaces with the Weil-Petersson distance function. A set of similarities the resulting metric structure shares with Thurston's asymmetric metric is noted.
We find bounds for Weil-Petersson holomorphic sectional curvature, and the Weil-Petersson curvature operator in several regimes, that do not depend on the topology of the underlying surface. Among other results, we show that the minimal (most negative) eigenvalue of the curvature operator at any point in the Teichmüller space $\Teich(S_g)$ of a closed surface $S_g$ of genus $g$ is uniformly bounded away from zero. Restricting to a thick part of $\Teich(S_g)$, we show that the minimal eigenvalue is uniformly bounded below by an explicit constant which does not depend on the topology of the surf…
In this paper we construct examples of Weil-Petersson geodesics with nonminimal ending laminations which have 1-dimensional limit sets in the Thurston compactification of Teichmüller space.
We prove that any mapping torus of a pseudo-Anosov mapping class with bounded normalized Weil-Petersson translation length contains a finite set of transverse and level closed curves, and drilling out this set of curves results in one of a finite number of cusped hyperbolic 3-manifolds. The number of manifolds in the finite list depends only on the bound for normalized translation length. We also prove a complementary result that explains the necessity of removing level curves by producing new estimates for the Weil-Petersson translation length of compositions of pseudo-Anosov mapping classes …
To a complex projective structure $Σ$ on a surface, Thurston associates a locally convex pleated surface. We derive bounds on the geometry of both in terms of the norms $\|ϕ_Σ\|_\infty$ and $\|ϕ_Σ\|_2$ of the quadratic differential $ϕ_Σ$ of $Σ$ given by the Schwarzian derivative of the associated locally univalent map. We show that these give a unifying approach that generalizes a number of important, well known results for convex cocompact hyperbolic structures on 3-manifolds, including bounds on the Lipschitz constant for the nearest-point retraction and the length of the bending lamination.…
We prove that the geodesic flow for the Weil-Petersson metric on the moduli space of Riemann surfaces is ergodic (in fact Bernoulli) and has finite, positive metric entropy.
Thanks to a theorem of Brock on comparison of Weil-Petersson translation distances and hyperbolic volumes of mapping tori for pseudo-Anosovs, we prove that the entropy of a surface automorphism in general has linear bounds in terms of Gromov norm of its mapping torus from below and in bounded geometry case from above. We also prove that the Weil-Petersson translation distance does the same from both sides in general. The proofs are in fact immediately derived from the theorem of Brock together with some other strong theorems and small observations.
Let S be a non-exceptional oriented surface of finite type. We discuss the action of subgroups of the mapping class group of S on the CAT(0)-boundary of the completion of Teichmueller space with respect to the Weil-Petersson metric. We show that the set of invariant Borel probability measures for the Weil-Petersson flow on moduli space which are supported on closed orbits is dense in the space of invariant Borel probability measures.
Moduli spaces of hyperbolic surfaces may be endowed with a symplectic structure via the Weil-Petersson form. Mirzakhani proved that Weil-Petersson volumes exhibit polynomial behaviour and that their coefficients store intersection numbers on moduli spaces of curves. In this survey article, we discuss these results as well as some consequences and applications.
Moduli spaces of hyperbolic surfaces with geodesic boundary components of fixed lengths may be endowed with a symplectic structure via the Weil-Petersson form. We show that, as the boundary lengths are sent to infinity, the Weil-Petersson form converges to a piecewise linear form first defined by Kontsevich. The proof rests on the observation that a hyperbolic surface with large boundary lengths resembles a graph after appropriately scaling the hyperbolic metric. We also include some applications to intersection theory on moduli spaces of curves.
We construct Weil-Petersson (WP) geodesic rays with minimal filling non-uniquely ergodic ending lamination which are recurrent to a compact subset of the moduli space of Riemann surfaces. This construction shows that an analogue of the Masur's criterion for Teichmüller geodesics does not hold for WP geodesics.
We introduce a coarse combinatorial description of the Weil-Petersson distance d_WP(X,Y) between two finite area hyperbolic Riemann surfaces X and Y. The combinatorics reveal a connection between Riemann surfaces and hyperbolic 3-manifolds conjectured by Thurston: the volume of the convex core of the quasi-Fuchsian manifold Q(X,Y) with X and Y in its boundary is comparable to the Weil-Petersson distance d_WP(X,Y). Applications include a connection of the Weil-Petersson distance with the Hausdorff dimension of the limit set and the lowest eigenvalue of the Laplacian as well as a new finiteness …
We define an ending lamination for a Weil-Petersson geodesic ray. Despite the lack of a natural visual boundary for the Weil-Petersson metric, these ending laminations provide an effective boundary theory that encodes much of its asymptotic CAT(0) geometry. In particular, we prove an ending lamination theorem (Theorem 1.1) for the full-measure set of rays that recur to the thick part, and we show that the association of an ending lamination embeds asymptote classes of recurrent rays into the Gromov-boundary of the curve complex. As an application, we establish fundamentals of the topological d…
For any pseudo-Anosov automorphism on an orientable closed surface, an inquality is established bounding certain growth of virtual homological eigenvalues with the Weil--Petersson translation length. The new inquality fits nicely with other known inequalities due to Kojima and McShane, and due to Lê. The new quantity to be considered is the square sum of the logarithmic radii of the homological eigenvalues (with multiplicity) outside the complex unit circle, called the homological Jensen square sum. The main theorem is as follows. For any cofinal sequence of regular finite covers of a given su…
We give an overview of the proof for Mirzakhani's volume recursion for the Weil-Petersson volumes of the moduli spaces of genus $g$ hyperbolic surfaces with $n$ labeled geodesic boundary components, and her application of this recursion to Witten's conjecture and the study of simple geodesic length spectrum growth rates.
We survey the renormalized volume of hyperbolic 3-manifolds, as a tool for Teichmuller theory, using simple differential geometry arguments to recover results sometimes first achieved by other means. One such application is McMullen's quasifuchsian (or more generally Kleinian) reciprocity, for which different arguments are proposed. Another is the fact that the renormalized volume of quasifuchsian (or more generally geometrically finite) hyperbolic 3-manifolds provides a Kahler potential for the Weil-Petersson metric on Teichmuller space. Yet another is the fact that the grafting map is symple…
A recent preprint of S. Kojima and G. McShane [KM] observes a beautiful explicit connection between Teichmüller translation distance and hyperbolic volume. It relies on a key estimate which we supply here: using geometric inflexibility of hyperbolic 3-manifolds, we show that for $S$ a closed surface, and $ψ\in \text{Mod}(S)$ pseudo-Anosov, the double iteration $Q(ψ^{-n}(X),ψ^n(X))$ has convex core volume differing from $2n \text{vol}(M_ψ)$ by a uniform additive constant, where $M_ψ$ is the hyperbolic mapping torus for $ψ$. We combine this estimate with work of Schlenker, and a branched coverin…
The space of broken hyperbolic structures generalizes the Teichmüller space of a punctured surface, and the space of projectivized broken measured foliations (equivalently, the space of projectivized affine foliations) generalizes the space of projectivized measured foliations. Just as projectivized measured foliations provide Thurston's boundary for Teichmüller space, so too do projectivized broken measured foliations form a boundary for the space of broken hyperbolic structures. In this paper, we naturally extend the Weil-Petersson Kähler two-form and the Thurston symplectic form to their br…
Maximal discs of Weil-Petersson class in $\mathbb{A}\mathrm{d}\mathbb{S}^{2,1}$
研究Weil-Petersson类在3维反de Sitter空间中的最大圆盘，证明Mess映射在Weil-Petersson万有Teichmüller空间的余切 bundle 上的辛微分同胚，并展示反共轭能量函数作为Kähler势的作用。
We analyze the coarse geometry of the Weil-Petersson metric on Teichmüller space, focusing on applications to its synthetic geometry (in particular the behavior of geodesics). We settle the question of the strong relative hyperbolicity of the Weil-Petersson metric via consideration of its coarse quasi-isometric model, the "pants graph." We show that in dimension~3 the pants graph is strongly relatively hyperbolic with respect to naturally defined product regions and show any quasi-flat lies a bounded distance from a single product. For all higher dimensions there is no non-trivial collection o…
Systole functions and Weil-Petersson geometry
本文研究Teichmüller空间中沿systolic曲线的测地线长度函数梯度的L^p范数，证明其与systole的幂次一致可比，并由此得出最小Weil-Petersson全纯截面曲率有统一负上界，否定了Mirzakhani的问题。
Hexagon decompositions and the Weil-Petersson metric
本文构造了从六边形图到增广Teichmüller空间的拟等距映射，并证明其宽度至多$\sqrt{g+n}\log(g+n)$，同时给出边界层的显式投影。
We consider the Weil--Petersson (WP) metric on the modular surface. We lift WP geodesics to the universal cover of the modular surface and analyse geometric properties of the lifts as paths in the hyperbolic metric on the universal cover. For any pair of distinct points in the thick part of the universal cover, we prove that the WP and hyperbolic geodesic segments that connect the pair, fellow-travel in the thick part and all deviations between these segments arise during cusp excursions. Furthermore, we give a quantitative analysis of the deviation during an excursion. We leverage the fellow …
A topological proof of Wolpert's formula for the Weil-Petersson symplectic form in terms of the Fenchel-Nielsen coordinates
本文引入与裤子分解相关的闭定向曲面自然胞腔分解及表示 Teichmüller 空间点的标准上闭链，并借此给出用 Fenchel-Nielsen 坐标表示 Weil-Petersson 辛形式的 Wolpert 公式的拓扑证明。
Analytic Theory on the Space of Blaschke Products: Simultaneous Uniformization and Weil-Petersson Metric
本文研究Blaschke积模空间的复结构及压力度量，证明了同时统一化定理并展示了压力半范数在超吸引子外的非退化性。
Critical Exponents on Hyperbolic Surfaces with Long Boundaries and the Asymptotic Weil-Petersson Form
该研究以归一化Weil-Petersson测度为概率测度，利用脊图构造对比双曲曲面与度量缎带图模空间的临界指数随机变量，证明长边界下的渐近均值收敛及测度拉回的定量一致估计。
A tale of two volumes of moduli spaces: Weil-Petersson and Masur-Veech
本文综述了Weil-Petersson与Masur-Veech体积的计算方法、关键结果及开放问题，探讨了两者在模空间研究中的联系与差异。
Infinite circle patterns in the Weil-Petersson class
本文研究了Weil-Petersson类中由离散调和函数参数化的无限圆图案空间，揭示了其与双曲体积泛函Hessian诱导的Riemann度量的关系。
Spectral gap with polynomial rate for Weil-Petersson random surfaces
该研究证明Weil-Petersson随机采样的亏格$g$闭双曲曲面拉普拉斯谱间隙至少为$\frac{1}{4}-O(\frac{1}{g^c})$（$g\to\infty$时概率趋近1），拓展并新证Anantharaman和Monk的结果，适配随机矩阵强收敛多项式方法至该场景。
A high-genus asymptotic expansion of Weil-Petersson volume polynomials
本文研究Weil-Petersson体积多项式在亏格趋于无穷时的渐近行为，利用Mirzakhani拓扑递归公式证明其按$g$负幂次展开，并显式计算第二项。
We study Weil-Petersson (WP) geodesics with narrow end invariant and develop techniques to control length-functions and twist parameters along them and prescribe their itinerary in the moduli space of Riemann surfaces. This class of geodesics is rich enough to provide for examples of closed WP geodesics in the thin part of the moduli space, as well as divergent WP geodesic rays with minimal filling ending lamination. Some ingredients of independent interest are the following: A strength version of Wolpert's Geodesic Limit Theorem proved in Sec.4. The stability of hierarchy resolution paths bet…
Local rigidity of covering constructions and Weil--Petersson subvarieties of the moduli space of curves
本文证明了曲线模空间$\u039c_{g,n}$的Weil–Petersson全测地子簇及覆盖构造是局部刚性的，该结论由一类轨形映射的一般刚性结果推导得到。
The Weil-Petersson gradient flow of renormalized volume and 3-dimensional convex cores
本文利用重正规化体积的 Weil-Petersson 梯度流研究相对无柱三维流形的双曲结构空间，引入手术流处理退化并证明重正规化体积与 WP 距离的线性下界，统一推广了多个双曲三维流形经典定理。
Large-$n$ asymptotics for Weil-Petersson volumes of moduli spaces of bordered hyperbolic surfaces
研究大$n$极限下Weil-Petersson随机曲面的谱理论，证明小Laplace特征值个数与$n$线性相关且最优，并给出模空间Weil-Petersson体积的新渐近公式。
Topological recursion of the Weil-Petersson volumes of hyperbolic surfaces with tight boundaries
本文将 Mirzakhani 的 Weil-Petersson 体积递归推广到具有紧特殊测地边界和锥形缺陷的双曲曲面，证明体积及生成函数仍为多项式并满足广义拓扑递归，进而用紧 Weil-Petersson 生成函数表示带缺陷的 JT 引力多边界关联函数。
We consider maps on a surface of genus $g$ with all vertices of degree at least three and positive real lengths assigned to the edges. In particular, we study the family of such metric maps with fixed genus $g$ and fixed number $n$ of faces with circumferences $α_1,\ldots,α_n$ and a $β$-irreducibility constraint, which roughly requires that all contractible cycles have length at least $β$. Using recent results on the enumeration of discrete maps with an irreducibility constraint, we compute the volume $V_{g,n}^{(β)}(α_1,\ldots,α_n)$ of this family of maps that arises naturally from the Lebesgu…
In this paper we consider two piecewise Riemannian metrics defined on the Culler-Vogtmann outer space which we call the entropy metric and the pressure metric. As a result of work of McMullen, these metrics can be seen as analogs of the Weil-Petersson metric on the Teichmüller space of a closed surface. We show that while the geometric analysis of these metrics is similar to that of the Weil-Petersson metric, from the point of view of geometric group theory, these metrics behave very differently to the Weil-Petersson metric. Specifically, we show that when the rank $r$ is at least 4, the actio…
A tree bijection for the moduli space of genus-0 hyperbolic surfaces with boundaries
本文通过树映射证明了具有边界的亏格为0的双曲表面的Weil-Petersson体积为边界长度的多项式函数，并建立了距离依赖的三点函数公式。
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