Linear structures on measured geodesic laminations

Laminations géodésiques plates mesurées

In [Mor], we have introduced a notion of flat laminations on surfaces endowed with a flat structure, similar to geodesic laminations on hyperbolic surfaces. Here is a sequel to this article that aims at defining transversal measures on flat laminations similar to transversal measures on hyperbolic laminations, taking into account that two different leaves of a flat lamination may no longer disjoint. Then, we define a topology on the set of measured flat laminations and show that its projective space is compact. Finally, we define the dual tree to a measured flat lamination.

Thurston 范数、$L^2$-范数、测地层状结构与 Lipschitz 映射

Thurston norms, $L^2$-norms, geodesic laminations, and Lipschitz maps

本文研究闭双曲 3-流形中 $L^2$-范数与 Thurston 范数的尖锐不等式，并结合最佳 Lipschitz 圆值映射和极大拉伸层状结构揭示熵与几何长度的关系。

Projective structures, grafting, and measured laminations

We show that grafting any fixed hyperbolic surface defines a homeomorphism from the space of measured laminations to Teichmuller space, complementing a result of Scannell-Wolf on grafting by a fixed lamination. This result is used to study the relationship between the complex-analytic and geometric coordinate systems for the space of complex projective ($\CP^1$) structures on a surface. We also study the rays in Teichmuller space associated to the grafting coordinates, obtaining estimates for extremal and hyperbolic length functions and their derivatives along these grafting rays.

凸余紧双曲流形由其褶皱层合唯一确定

Convex co-compact hyperbolic manifolds are determined by their pleating lamination

本文证明凸余紧三维双曲流形可由其凸核边界上的褶皱测度层合唯一确定，解决了该类流形的刚性问题。

Stretch 映射与测地叶状层的解析性质

Analytic properties of Stretch maps and geodesic laminations

本文通过 p-Schatten 调和映射在 p→∞ 时的极限构造最佳 Lipschitz 映射，并证明极限对偶测度 dv 的支集位于 Thurston 典范测地叶状层上。

Fixed points of compositions of earthquakes

Let S be a closed surface of genus at least 2, and consider two measured geodesic laminations that fill S. Right earthquakes along these laminations are diffeomorphisms of the Teichmüller space of S. We prove that the composition of these earthquakes has a fixed point in the Teichmüller space. Another way to state this result is that it is possible to prescribe any two measured laminations that fill a surface as the upper and lower measured bending laminations of the convex core of a globally hyperbolic AdS manifold. The proof uses some estimates from the geometry of those AdS manifolds.

Earthquakes and graftings of hyperbolic surface laminations

We study compact hyperbolic surface laminations. These are a generalization of closed hyperbolic surfaces which appear to be more suited to the study of Teichmüller theory than arbitrary non-compact surfaces. We show that the Teichmüller space of any non-trivial hyperbolic surface lamination is infinite dimensional. In order to prove this result, we study the theory of deformations of hyperbolic surfaces, and we derive what we believe to be a new formula for the derivative of the length of a simple closed geodesic with respect to the action of grafting. This formula complements those derived b…

Grafting, pruning, and the antipodal map on measured laminations

Grafting a measured lamination on a hyperbolic surface defines a self-map of Teichmuller space, which is a homeomorphism by a result of Scannell and Wolf. In this paper we study the large-scale behavior of pruning, which is the inverse of grafting. Specifically, for each conformal structure $X \in \T(S)$, pruning $X$ gives a map $\ML(S) \to \T(S)$. We show that this map extends to the Thurston compactification of $\T(S)$, and that its boundary values are the natural antipodal involution relative to $X$ on the space of projective measured laminations. We use this result to study Thurston's graf…

Variations of the boundary geometry of 3--dimensional hyperbolic convex cores

A fundamental object in a hyperbolic 3-manifold M is its convex core C(M), defined as the smallest closed non-empty convex subset of M. We investigate the way the geometry of the boundary S of C(M) varies as we vary the hyperbolic metric of M. Thurston observed that the intrinsic metric of S is hyperbolic, and that its bending is described by a measured geodesic lamination. We show that, as the hyperbolic metric of the 3--manifold M varies differentiably, the hyperbolic metric of the surface S varies in a C^1, but usually not C^2, manner. Differentiability properties for the bending measured l…

The Schwarzian derivative and measured laminations on Riemann surfaces

We compare two relationships between quadratic differentials and measured geodesic laminations on hyperbolic Riemann surfaces (by foliations or complex projective structures). Each yields a homeomorphism $\ML(S) \to Q(X)$ for any conformal structure $X$ on a compact surface $S$. The main result is that these maps are nearly the same, differing by a multiplicative factor of -2 and an error term of lower order than the maps themselves (which we bound explicitly). As an application we show that the Schwarzian derivative of a $\CP^1$ structure with Fuchsian holonomy is close to a $2π$-integral Jen…

Minimal stretch maps between hyperbolic surfaces

This paper develops a theory of Lipschitz comparisons of hyperbolic surfaces analogous to the theory of quasi-conformal comparisons. Extremal Lipschitz maps (minimal stretch maps) and geodesics for the `Lipschitz metric' are constructed. The extremal Lipschitz constant equals the maximum ratio of lengths of measured laminations, which is attained with probability one on a simple closed curve. Cataclysms are introduced, generalizing earthquakes by permitting more violent shearing in both directions along a fault. Cataclysms provide useful coordinates for Teichmuller space that are convenient fo…

Hamiltonian properties of earthquakes flows on surfaces with closed geodesic boundary

The Teichmüller space $\mathcal{T}_S(\mathbf{b})$ of hyperbolic metrics on a surface $S$ with fixed lengths at the boundary components is symplectic. We prove that any sum of infinitesimal earthquakes on $S$ that is tangent to $\mathcal{T}_S(\mathbf{b})$ is Hamiltonian, by providing a Hamiltonian $\mathbb{L}$. Such function extends the classical length map associated to a compactly supported measured geodesic lamination and shares with it some peculiar properties, such as properness and strict convexity along earthquakes paths under usual topological conditions. As an application, we prove tha…

2 π-grafting and complex projective structures, I

Let $S$ be a closed oriented surface of genus at least two. Gallo, Kapovich, and Marden asked if 2π-graftings produce all projective structures on $S$ with arbitrarily fixed holonomy (Grafting Conjecture). In this paper, we show that the conjecture holds true "locally" in the space $GL$ of geodesic laminations on $S$ via a natural projection of projective structures on $S$ into $GL$ in the Thurston coordinates. In the sequel paper, using this local solution, we prove the conjecture for generic holonomy.

Notes on the Schwarzian tensor and measured foliations at infinity of quasifuchsian manifolds

The boundary at infinity of a quasifuchsian hyperbolic manifold is equiped with a holomorphic quadratic differential. Its horizontal measured foliation $f$ can be interpreted as the natural analog of the measured bending lamination on the boundary of the convex core. This analogy leads to a number of questions. We provide a variation formula for the renormalized volume in terms of the extremal length $\ext(f)$ of $f$, and an upper bound on $\ext(f)$. We then describe two extensions of the holomorphic quadratic differential at infinity, both valid in higher dimensions. One is in terms of Poinca…