Meromorphic quadratic differentials and measured foliations on a Riemann surface

Meromorphic quadratic differentials with complex residues and spiralling foliations

A meromorphic quadratic differential with poles of order two, on a compact Riemann surface, induces a measured foliation on the surface, with a spiralling structure at any pole that is determined by the complex residue of the differential at the pole. We introduce the space of such measured foliations, and prove that for a fixed Riemann surface, any such foliation is realized by a quadratic differential with second order poles at marked points. Furthermore, such a differential is uniquely determined if one prescribes complex residues at the poles that are compatible with the transverse measure…

Quadratic differentials, measured foliations and metric graphs on punctured surfaces

A meromorphic quadratic differential on a punctured Riemann surface induces horizontal and vertical measured foliations with pole-singularities. In a neighborhood of a pole such a foliation comprises foliated strips and half-planes, and its leaf-space determines a metric graph. We introduce the notion of an asymptotic direction at each pole, and show that for a punctured surface equipped with a choice of such asymptotic data, any compatible pair of measured foliations uniquely determines a complex structure and a meromorphic quadratic differential realizing that pair. This proves the analogue …

亚纯联络在具有不规则奇点时的仿射几何

The affine geometry of meromorphic connections with irregular singularities

本文研究黎曼曲面切丛上亚纯联络的仿射几何，引入局部不变量和局部模型，推广Veech构造定义Delaunay分解，并给出复杂度上界。

Meromorphic $L^2$ functions on flat surfaces

We prove a quantitative version of the non-uniform hyperbolicity of the Teichmüller geodesic flow. Namely, at each point of any Teichmüller flow line, we bound the infinitesimal spectral gap for variations of the Hodge norm along the flow line in terms of an easily estimated geometric quantity on the flat surface, which is greater than or equal to the flat systole. As applications, we strengthen results of Treviño and Smith regarding unique ergodicity of measured foliations, and give an estimate for the spectral gaps of pseudo-Anosov homeomorphisms based on the location of their axes in the mo…