Harmonic branched coverings and uniformization of CAT($k$) spheres

Discrete uniformization of finite branched covers over the Riemann sphere via hyper-ideal circle patterns

With the help of hyper-ideal circle pattern theory, we have developed a discrete version of the classical uniformization theorems for surfaces represented as finite branched covers over the Riemann sphere as well as compact polyhedral surfaces with non-positive curvature. We show that in the case of such surfaces discrete uniformization via hyper-ideal circle patterns always exists and is unique. We also propose a numerical algorithm, utilizing convex optimization, that constructs the desired discrete uniformization.

Uniformization of branched surfaces and Higgs bundles

Given a compact Riemann surface $Σ$ of genus $g_Σ\, \geq\, 2$, and an effective divisor $D\, =\, \sum_i n_i x_i$ on $Σ$ with $\text{degree}(D)\, <\, 2(g_Σ-1)$, there is a unique cone metric on $Σ$ of constant negative curvature $-4$ such that the cone angle at each $x_i$ is $2πn_i$ (see McOwen and Troyanov [McO,Tr]). We describe the Higgs bundle corresponding to this uniformization associated to the above conical metric. We also give a family of Higgs bundles on $Σ$ parametrized by a nonempty open subset of $H^0(Σ,\,K_Σ^{\otimes 2}\otimes{\mathcal O}_Σ(-2D))$ that correspond to conical metrics…

Bers同时单值化与庞加莱和乐簇的交

Bers' simultaneous uniformization and the intersection of Poincare holonomy varieties

本文研究具有相同和乐的不同$\u2102P^1$-结构有序对空间，证明其到泰希米勒空间乘积去对角线的映射是完备局部分歧覆盖，重证Bers同时单值化定理，并得出任意两个庞加莱和乐簇的交为非空离散集的结论。

Metric curvature of infinite branched covers

We study branched covering spaces in several contexts, proving that under suitable circumstances the cover satisfies the same upper curvature bounds as the base space. The first context is of a branched cover of an arbitrary metric space that satisfies Alexandrov's curvature condition CAT(k), over an arbitrary complete convex subset. The second context is of a certain sort of branched cover of a Riemannian manifold over a family of mutually orthogonal submanifolds. In neither setting do we require that the branching be locally finite. We apply our results to hyperplane complements in several c…

稳定抛物Higgs丛的复代数簇轨道均匀化

Orbifold Uniformization of Complex Algebraic Variety via Polystable Parabolic Higgs Bundle

研究带有简单正规交叉除子的光滑复射影簇上的抛物Higgs丛，通过抛物Bogomolov-Gieseker不等式等号情形，得到分支复双曲结构，并给出标准情形下的无分歧轨道球均匀化。

格罗莫夫-瑟斯顿流形与分支覆盖的熵

The entropy of Gromov-Thurston manifolds and branched coverings

该研究建立局部CAT(k)分支覆盖测地流动力学的一般理论，推导其熵的显式公式，得到熵随分支覆盖片数趋于无穷的渐近及最大熵测度的渐近性质。