Given a harmonic measure of a hyperbolic lamination on a compact metric space, a positive harmonic function is defined on the universal cover of a typical leaves. We discuss some properties of this function. Especially if all the leaves are hyperbolic, ergodic harmonic measures are divided into two classes.
We construct minimal laminations by hyperbolic surfaces whose generic leaf is a disk and contain any prescribed family of surfaces and with a precise control of the topologies of the surfaces that appear. The laminations are constructed via towers of finite coverings of surfaces for which we need to develop a relative version of residual finiteness which may be of independent interest. The main step in establishing this relative version of residual finiteness is to obtain finite covers with control on the \emph{second systole} of the surface, which is done in the appendix. In a companion paper…
We give the topological obstructions to be a leaf in a minimal lamination by hyperbolic surfaces whose generic leaf is homeomorphic to a Cantor tree. Then, we show that all allowed topological types can be simultaneously embedded in the same lamination. This result, together with results of Alvarez-Brum-Martínez-Potrie and Blanc, complete the panorama of understanding which topological surfaces can be leaves in minimal hyperbolic surface laminations when the topology of the generic leaf is given. In all cases, all possible topologies can be realized simultaneously.
Thurston and Calegari-Dunfield showed that the fundamental group of some tautly foliated hyperbolic 3-manifold acts on the circle in a distinctive way that the action preserves some structure of S^1, so-called a circle lamination. Indeed, a large class of Kleinian groups acts on the circle preserving a circle lamination. In this paper, we are concerned with the converse problem that a group acting on the circle with at least two invariant circle laminations is Kleinian. We prove that a subgroup of the orientation preserving circle homeomorphism group is a Fuchsian group whose quotient orbifold…
Let Gamma_0 be a discrete group. For a pair (j,rho) of representations of Gamma_0 into PO(n,1)=Isom(H^n) with j geometrically finite, we study the set of (j,rho)-equivariant Lipschitz maps from the real hyperbolic space H^n to itself that have minimal Lipschitz constant. Our main result is the existence of a geodesic lamination that is "maximally stretched" by all such maps when the minimal constant is at least 1. As an application, we generalize two-dimensional results and constructions of Thurston and extend his asymmetric metric on Teichmüller space to a geometrically finite setting and to …
Let $λ_-$ and $λ_+$ be two bounded measured laminations on the hyperbolic disk $\mathbb H^2$, which "strongly fill" (definition below). We consider the left earthquakes along $λ_-$ and $λ_+$, considered as maps from the universal Teichmüller space $\mathcal T$ to itself, and we prove that the composition of those left earthquakes has a fixed point. The proof uses anti-de Sitter geometry. Given a quasi-symmetric homeomorphism $u:{\mathbb RP}^1\to {\mathbb RP}^1$, the boundary of the convex hull in $AdS^3$ of its graph in ${\mathbb RP}^1\times{\mathbb RP}^1\simeq \partial AdS^3$ is the disjoint …
We introduce an asymmetric distance function, which we call the `left Hausdorff distance function', on the space of geodesic laminations on a closed hyperbolic surface of genus at least 2. This distance is an asymmetric version of the Hausdorff distance between compact subsets of a metric space. We prove a rigidity result for the action of the extended mapping class group of the surface on the space of geodesic laminations equipped with the topology induced from this distance. More specifically, we prove that there is a natural homomorphism from the extended mapping class group into the group …
In the present paper, we revisit a famous theorem by Candel that we generalize by proving that given a compact lamination by hyperbolic surfaces, every negative function smooth inside the leaves and transversally continuous is the curvature function of a unique laminated metric in the corresponding conformal class. We give an interpretation of this result as a continuity result about the solutions of some elliptic PDEs in the so called Cheeger-Gromov topology on the space of complete pointed riemannian manifolds.
Given integers $g,n \geq 0$ satisfying $2-2g-n < 0$, let $\mathcal{M}_{g,n}$ be the moduli space of connected, oriented, complete, finite area hyperbolic surfaces of genus $g$ with $n$ cusps. We study the global behavior of the Mirzakhani function $B \colon \mathcal{M}_{g,n} \to \mathbf{R}_{\geq 0}$ which assigns to $X \in \mathcal{M}_{g,n}$ the Thurston measure of the set of measured geodesic laminations on $X$ of hyperbolic length $\leq 1$. We improve bounds of Mirzakhani describing the behavior of this function near the cusp of $\mathcal{M}_{g,n}$ and deduce that $B$ is square-integrable wi…
Given a connected, oriented, complete, finite area hyperbolic surface $X$ of genus $g$ with $n$ punctures, Mirzakhani showed that the number of multi-geodesics on $X$ of total hyperbolic length $\leq L$ in the mapping class group orbit of a given simple or filling closed multi-curve is asymptotic as $L \to \infty$ to a polynomial in $L$ of degree $6g-6+2n$. We establish asymptotics of the same kind for countings of multi-geodesics in mapping class group orbits of simple or filling closed multi-curves that keep track of the hyperbolic lengths of individual components, proving and generalizing a…
For every half-translation surface with marked points $(M,Σ)$, we construct an associated tessellation $Π(M,Σ)$ of the Poincaré upper half plane whose tiles have finitely many sides and area at most $π$. The tessellation $Π(M,Σ)$ is equivariant with respect to the action of $\mathrm{PSL}(2,\mathbb{R})$, and invariant with respect to (half-)translation covering. In the case $(M,Σ)$ is the torus $\mathbb{C}/\mathbb{Z}^2$ with a one marked point, $Π(\mathbb{C}/\mathbb{Z}^2,\{0\})$ coincides with the iso-Delaunay tessellation introduced by Veech as both tessellations give the Farey tessellation. A…
We consider harmonic diffeomorphisms to a fixed hyperbolic target $Y$, from a family of domain Riemann surfaces degenerating along a Teichmüller ray. We use the work of Minsky to show that there is a limiting harmonic map from the conformal limit of the Teichmüller ray, to a crowned hyperbolic surface. The target surface is the metric completion of the complement of a geodesic lamination on $Y$. The conformal limit is obtained by attaching half-planes and cylinders to the critical graph of the holomorphic quadratic differential determining the ray. As an application, we provide a new proof of …
For a given pseudo-Anosov homeomorphism $φ$ of a closed surface $S$, the action of $φ$ on the Teichmüller space $\mathcal T(S)$ preserves the Weil-Petersson symplectic form. We give explicit formulae for two invariant functions $\mathcal T(S)\to \mathbb R$ whose symplectic gradients generate autonomous Hamiltonian flows that coincide with the action of $φ$ at time one. We compute the Poisson bracket between these two functions. This amounts to computing the variation of length of a Hölder cocyle on one lamination along a shear vector field defined by another. For a measurably generic set of la…
Limit points of lines of minima in Thurston's boundary of Teichmueller space
给定在双曲曲面中填充该曲面的两个测量的叠层μ和nu，Kerckhoff [Lines of Minima in Teichmueller space，Duke Math J.65（1992）187-213]定义了一条相关的最小线，沿着该最小线，μ和nu的长度函数的凸组合被最小化。这是Teichmueller空间中的一条线，可以被认为类似于双曲空间中由无穷远处的两点确定的测地线。我们表明，当亩是唯一遍历，这条线收敛到投影层[亩]，但当亩是合理的，这条线不收敛到[亩]，而是重心的支持亩。Masur [Two boundaries of Teichmueller space，Duke Math.J.49（1982）183-190]证明了关于Teichmueller测地线行为的类似结果。
