Workshop on Geometry and Analysis on singular spaces

摘要：

In this talk, we are interested in a class of GHMC (globally hyperbolic maximal compact) \((2+1)\)-spacetimes of constant curvature with particles (cone singularities of angles less than π along time-like lines). We will discuss the existence and uniqueness of two special foliations of those spacetimes, by space-like surfaces of constant Gauss curvature (called K-foliation) and by space-like surfaces of constant mean curvature (called CMC foliation). We will also explain their connection with Teichmüller theory of hyperbolic surfaces with cone singularities. Those results are based on some joint-work with Jean-Marc Schlenker and Andrea Tamburelli.

摘要：

In this talk, I will introduce some recent results on the spectrum of the classical Lame operator. Some applications to the mean field equation with singularities on flat tori will also be given.

摘要：

Every half-translation surface induces a singular flat metric. A saddle connection is an open geodesic segment which connects singular points and which contains no singular points in the interior. The saddle connection graph is a graph whose vertices are saddle connections and edges are pairs of disjoint saddle connections. In this talk, we will discuss the rigidity and the coarse geometry of saddle connection graphs. We will show that on the one hand two saddle connection graphs are isometric if and only if the underlying half-translation surfaces are affine equivalent, while on the other hand all saddle connections graphs are uniformly quasi-isometric to the regular infinite-valency tree. This talk is partially based on a joint work with Valentina Disarlo, Anja Rendecker and Robert Tang.

摘要：

A cone spherical metric on a compact Riemann surface \(X\) is a conformal metric of Gaussian curvature \(+1\) with finitely many conical singularities. The singularities of the metric can be described by a real divisor \(D\) with coefficients \(> -1\). An open question is whether there exists a cone spherical metric on \(X\) for properly given \((X,D)\) such that the singularities of the metric are described by \(D\). In this talk, by using projective structures, I will give a correspondence between cone spherical metrics representing effective divisors and extensions of line bundles on Riemann surfaces. In particular, for any given stable bundle of rank two with a line subbundle, we could construct an irreducible metric. Then we define a real analytic map which is called ramification divisor map, and show that the existence and uniqueness problem of such metric is boiled down to understanding the corresponding ramification divisor map. This is a joint work with Lingguang Li and Bin Xu.

摘要：

In this talk, we will introduce the \(\sigma_2\) Yamabe problem on conic manifolds and give the necessary algebraic condition for the existence of solution in positive cone. We also prove a convergence theorem on the moduli space of constant \(\sigma_2\) metrics for conic 4-spheres. And in the negative cone, we consider conformal metrics on a unit 4-disc with an asymptotically hyperbolic end and possible isolated conic singularities. Under a positive \(\sigma_{2}\) curvature lower bound condition, we define a corresponding mass term and establish a Penrose-type inequality. We point out that our curvature condition is different from the scalar curvature condition required in earlier literature. This is joint work with Hao Fang.

摘要：

In this talk, we will consider Ricci soliton. The first result is the classification of kahler shrinking gradient Ricci soliton with nonnegative bisectional curvature, which is joint with Prof Shijin Zhang. Then I focus on the splitting results for soliton. If the integral of Ricci curvature along the geodesic line is nonnegative, then the splitting result hold for shrinking soliton. Under some weak pinching condition on the Ricci curvature, we obtain the shrinking soliton has only one end. Similar results hold for expanding soliton.

摘要：

Usually we call a one-dimensional non-CSC extremal Kähler metric HCMU(the Hessian of the Curvature of the Metric is Umbilical) metric. In this talk, I will first give the definition of HCMU metric. Then some properties and the basic equations of HCMU metric will be presented. Using the basic equations we will talk about the existence problem and the isometric immersion problem.

摘要：

In this talk, we will present recent progress on singular metrics with special curvatures, including Kahler-Einstein metrics, constant scalar curvature Kahler metrics, extremal metrics, etc.

摘要：

在这个报告中，我们首先简要回顾一下带锥性奇点的紧黎曼曲面的定义以及Gauss-Bonnet公式，其上的Sobolev空间及嵌入定理、Poincare不等式、极大值原理、上下解原理等分析知识。然后介绍一下杨云雁教授和我的新近的推广丁老师和刘嘉荃老师的光滑情形的结果。

摘要：

The problem of finding and classifying constant curvature metrics with conical singularities has a long history bringing together several different areas of mathematics. This talk will focus on the particularly difficult spherical case where many new phenomena appear. When some of the cone angles are bigger than \(2\pi\), uniqueness fails and existence is not guaranteed; smooth deformation is not always possible and the moduli space is expected to have singular strata. I will give a survey of several recent results regarding this singular uniformization problem, connecting microlocal techniques with complex analysis and synthetic geometry. Based on joint works with Rafe Mazzeo, Bin Xu, and Mikhail Karpukhin.