Multi Black Holes and Earthquakes on Riemann surfaces with boundaries

On the asymptotic behavior of complex earthquakes and Teichmüller disks

Given a hyperbolic surface and a simple closed geodesic on it, complex-twists along the curve produce a holomorphic family of deformations in Teichmüller space, degenerating to the Riemann surface where it is pinched. We show there is a corresponding Teichmüller disk such that the two are strongly asymptotic, in the Teichmüller metric, around the noded Riemann surface. We establish a similar comparison with plumbing deformations that open the node.

Teichmüller空间上的水平度量

The horocyclic metric on Teichm{ü}ller spaces

本文从共形视角研究Teichmüller空间上的水平度量，将Thurston的理论扩展到任意亏格的Riemann面，并通过复Legendre变换建立Weil-Petersson symplectic形式的对偶关系。

Disks area-minimizing in mean convex Riemannian $n$-manifolds

We prove the validity of an inequality involving a mean of the area and the length of the boundary of immersed disks whose boundaries are homotopically non-trivial curves in an oriented compact manifold which possesses convex mean curvature boundary, positive escalar curvature and admits a map to $\mathbb{D}^2\times T^{n}$ with nonzero degree, where $\mathbb{D}^2$ is a disk and $T^n$ is an $n$-dimensional torus. We also prove a rigidity result for the equality case when the boundary is totally geodesic. This can be viewed as a partial generalization of a result due to Lucas Ambrózio in \cite{A…

The Thurston metric on hyperbolic domains and boundaries of convex hulls

We show that the nearest point retraction is a uniform quasi-isometry from the Thurston metric on a hyperbolic domain in the Riemann sphere to the boundary of the convex hull of its complement. As a corollary, one obtains explicit bounds on the quasi-isometry constant of the nearest point retraction with respect to the Poincare metric when the domain is uniformly perfect. We also establish Marden and Markovic's conjecture that a hyperbolic domain is uniformly perfect if and only if the nearest point retraction is Lipschitz with respect to the Poincare metric.

A Thurston boundary for infinite-dimensional Teichmüller spaces

For a compact surface $X_0$, Thurston introduced a compactification of its Teichmüller space $\mathcal T(X_0)$ by completing it with a boundary $\mathcal{PML}(X_0)$ consisting of projective measured geodesic laminations. We introduce a similar bordification for the Teichmüller space $\mathcal T(X_0)$ of a noncompact Riemann surface $X_0$, using the technical tool of geodesic currents. The lack of compactness requires the introduction of certain uniformity conditions which were unnecessary for compact surfaces. A technical step, providing a convergence result for earthquake paths in $\mathcal T…

Quasicircle boundaries and exotic almost-isometries

We consider properly discontinuous, isometric, convex cocompact actions of surface groups on a CAT(-1) space. We show that the limit set of such an action, equipped with the canonical visual metric, is a (weak) quasicircle in the sense of Falconer and Marsh. It follows that the visual metrics on such limit sets are classified, up to bi-Lipschitz equivalence, by their Hausdorff dimension. This result applies in particular to boundaries at infinity of the universal cover of a locally CAT(-1) surface. We show that any two periodic CAT(-1) metrics on $\mathbb H^2$ can be scaled so as to be almost-…