Harmonic maps and wild Teichmüller spaces

在Goldman空间中Teichmüller空间是完全测地线的

Teichmüller Space Is Totally Geodesic In Goldman Space

本文在曲面$\mathcal{B}(S)$上的凸投射结构的等价类空间$\mathcal{B}(S)$上构造了一个新的黎曼度量，并证明了这个新的度量与Darvishzadeh和Goldman的度量一样，都是嵌入到Goldman空间$\mathcal{B}(S)$的子流形上的TeichmPDMATHABMOller空间上的Weil-Petersson度量.此外，赋予Weil-Petersson度量的$\ddot{u}$空间在黎曼流形$\mathcal{B}(S)$中是全测地的.

A note on Nicolas-Auguste Tissot: At the origin of quasiconformal mappings (to appear in Vol. VII of the Handbook of Teichm{ü}ller Theory)

Nicolas-Auguste Tissot (1824--1897) was a French mathematician and cartographer. He introduced a tool which became known among geographers under the name ``Tissot indicatrix'', and which was widely used during the first half of the twentieth century in cartography. This is a graphical representation of a field of ellipses, indicating at each point of a geographical map the distorsion of this map, both in direction and in magnitude. Each ellipse represented at a given point is the image of an infinitesimal circle in the domain of the map (generally speaking, a sphere representing the surface of…

Wild globally hyperbolic maximal anti-de Sitter structures

Let $Σ$ be a connected, oriented surface with punctures and negative Euler characteristic. We introduce wild globally hyperbolic anti-de Sitter structures on $Σ\times \mathbb{R}$ and provide two parameterisations of their deformation space: as a quotient of the product of two copies of the Teichmüller space of crowned hyperbolic surfaces and as the bundle over the Teichmüller space of $Σ$ of meromorphic quadratic differentials with poles of order at least $3$ at the punctures.

Plurisuperharmonicity of reciprocal energy function on Teichmuller space and Weil-Petersson metrics

We consider harmonic maps$u(z): \mathcal{X}_z\to N$ in a fixed homotopy class from Riemann surfaces $\mathcal{X}_z$ of genus $g\geq 2$ varying in the Teichmü{}ller space $\mathcal T$ to a Riemannian manifold $N$ with non-positive Hermitian sectional curvature. The energy function $E(z)=E(u(z))$ can be viewed as a function on $\mathcal T$ and we study its first and the second variations. We prove that the reciprocal energy function $E(z)^{-1}$ is plurisuperharmonic on Teichmüller space. We also obtain the (strict) plurisubharmonicity of $\log E(z)$ and $E(z)$. As an application, we get the foll…

Infinite energy equivariant harmonic maps, domination, and anti-de Sitter $3$-manifolds

We generalize a well-known existence and uniqueness result for equivariant harmonic maps due to Corlette, Donaldson, and Labourie to a non-compact infinite energy setting and analyze the asymptotic behaviour of the harmonic maps. When the relevant representation is Fuchsian and has hyperbolic monodromy, our construction recovers a family of harmonic maps originally studied by Wolf. We employ these maps to solve a domination problem for representations. In particular, following ideas laid out by Deroin-Tholozan, we prove that any representation from a finitely generated free group to the isomet…

Lorentzian Kleinian Groups

Classical Kleinian groups are discrete subgroups of isometries of H n. The well-known theory of Kleinian groups starts with the definition of their associated limit set in the boundary of H n , and includes the geometric properties of the quotient hyperbolic space. This approach, naively applied, fails in the Lorentzian analogue anti-de Sitter space: discrete subgroups do not act properly discontinuously, and in many cases the set of accumulation points of orbits at the conformal boundary at infinity depends on the orbit. In this survey, we point out a way to extend this classical theory by in…