Riemann surfaces with boundary and natural triangulations of the Teichmueller space

A scattering theory of harmonic one-forms on Riemann surfaces

We construct a scattering theory for harmonic one-forms on Riemann surfaces, obtained from boundary value problems through systems of curves and the jump problem. We obtain an explicit expression for the scattering matrix in terms of integral operators which we call Schiffer operators, and show that the matrix is unitary. As a consequence of this scattering theory, we prove index theorems relating these conformally invariant integral operators to topological invariants. We also obtain a general association of positive polarizing Lagrangian spaces to bordered Riemann surfaces, which unifies the…

Curvature based triangulation of metric measure spaces

We prove that a Ricci curvature based method of triangulation of compact Riemannian manifolds, due to Grove and Petersen, extends to the context of weighted Riemannian manifolds and more general metric measure spaces. In both cases the role of the lower bound on Ricci curvature is replaced by the curvature-dimension condition ${\rm CD}(K,N)$. We show also that for weighted Riemannian manifolds the triangulation can be improved to become a thick one and that, in consequence, such manifolds admit weight-sensitive quasimeromorphic mappings. An application of this last result to information manifo…

Mixing Flows on Moduli Spaces of Flat Bundles over Surfaces

We extend Teichmueller dynamics to a flow on the total space of a flat bundle of deformation spaces of representations of the fundamental group of a fixed surface S in a Lie group G. The resulting dynamical system is a continuous version of the action of the mapping class group of S on the deformation space. We observe how ergodic properties of this action relate to this flow. When G is compact, this flow is strongly mixing over each component of the derormation space and of each stratum of the Teichmueller unit sphere bundle over the Riemann moduli space. We prove ergodicity for the analogous…

Multi Black Holes and Earthquakes on Riemann surfaces with boundaries

We prove an "Earthquake Theorem" for hyperbolic metrics with geodesic boundary on a compact surfaces $S$ with boundary: given two hyperbolic metrics with geodesic boundary on a surface with $k$ boundary components, there are $2^k$ right earthquakes transforming the first in the second. An alternative formulation arises by introducing the enhanced Teichmueller space of S: We prove that any two points of the latter are related by a unique right earthquake. The proof rests on the geometry of ``multi-black holes'', which are 3-dimensional anti-de Sitter manifolds, topologically the product of a su…

没有准等距单纯 triangulations 的曲面

Surfaces without quasi-isometric simplicial triangulations

本文构造了一个无边界的完备黎曼曲面，该曲面不包含任何使得其单纯1骨架准等距嵌入的三角剖分，且曲面的最短环长可任意大，回答了Georgakopoulos的问题。

A Teichmuller space for negatively curved surfaces

We first describe the action of the fundamental group of a closed surface of variable negative curvature on the oriented geodesics in its universal covering in terms of a naturally-defined flat connection whose holonomy lies in the group of Hamiltonian diffeomorphisms of S^1 x R. Consideration of the holonomy necessitates an extension from Riemannian to Finsler metrics. The second part of the paper follows the Higgs bundle approach to flat connections adapted to this infinite dimensional group and focuses on a family of metrics, relying on a construction of O.Biquard, which is parametrized by …

Wall-crossing, Hitchin Systems, and the WKB Approximation

We consider BPS states in a large class of d=4, N=2 field theories, obtained by reducing six-dimensional (2,0) superconformal field theories on Riemann surfaces, with defect operators inserted at points of the Riemann surface. Further dimensional reduction on S^1 yields sigma models, whose target spaces are moduli spaces of Higgs bundles on Riemann surfaces with ramification. In the case where the Higgs bundles have rank 2, we construct canonical Darboux coordinate systems on their moduli spaces. These coordinate systems are related to one another by Poisson transformations associated to BPS s…

稳定性条件与泰希米勒空间

Stability conditions and Teichmüller space

本研究针对带标记有边曲面理想三角剖分对应的3-Calabi-Yau三角范畴，借助调和映射理论构建了从Bridgeland稳定性条件空间分支到增强泰希米勒空间的自然映射，并揭示了范畴对象中心荷与剪切坐标的关联。

模空间中三角剖分曲面分布的大属界

Large genus bounds for the distribution of triangulated surfaces in moduli space

针对三角剖分曲面在 Riemann 曲面模空间中的大属分布问题，本文通过估计 Teichmüller 球内曲面数量的上下界，证明其在强意义下分布良好，且单位球内数量至多随三角形数指数增长。

Bending Fuchsian representations of fundamental groups of cusped surfaces in PU(2,1)

We describe a family of representations of $π_1(Σ)$ in PU(2,1), where $Σ$ is a hyperbolic Riemann surface with at least one deleted point. This family is obtained by a bending process associated to an ideal triangulation of $Σ$. We give an explicit description of this family by describing a coordinates system in the spirit of shear coordinates on the Teichmüller space. We identify within this family new examples of discrete, faithful and type-preserving representations of $π_1(Σ)$. In turn, we obtain a 1-parameter family of embeddings of the Teichmüller space of $Σ$ in the PU(2,1)-representati…