Quillen metrics and branched coverings

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

Let $S$ be a surface with a metric $d$ satisfying an upper curvature bound in the sense of Alexandrov (i.e. via triangle comparison). We show that an almost conformal harmonic map from a surface into $(S,d)$ is a branched covering. As a consequence, if $(S,d)$ is homeomorphically equivalent to the 2-sphere $\mathbb S^2$, then it is conformally equivalent to $\mathbb S^2$.

Geometric Realizations of Cyclically Branched Coverings over Punctured Spheres

In classical differential geometry, a central question has been whether abstract surfaces with given geometric features can be realized as surfaces in Euclidean space. Inspired by the rich theory of embedded triply periodic minimal surfaces, we seek examples of triply periodic polyhedral surfaces that have an identifiable conformal structure. In particular, we are interested in explicit cone metrics on compact Riemann surfaces that have a realization as the quotient of a triply periodic polyhedral surface. This is important as Riemann surfaces where one has equivalent descriptions are rare. We…

Harmonic maps between surfaces homotopic to a (branched) covering map

In the paper, we consider the harmonic maps between surfaces $Σ$ and $S$ in the homotopy class of a (branched) covering map $u_0$. We prove the uniqueness of critical points of energy function and the injectivity of Hopf differential if $u_0$ is a covering map. On the other hand, if $u_0$ is a branched covering, we show that the uniqueness of critical points fails if $u_0$ is a non-simple branched covering, and prove the injectivity of Hopf differential $Φ:\mc{T}(S)\to \op{QD}(Σ,g)$ when $g=[u_0^* h]$ for some hyperbolic metric $h$ on $S$.

双曲流形的分支覆盖几何

The geometry of branched coverings of hyperbolic manifolds

研究双曲流形的分支覆盖几何性质，重点探讨高维闭合双曲流形沿完全测地子流形的覆盖结构及其几何特性。

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。

Immersed Lagrangian deformations of a branched covering of a special Lagrangian 3-sphere in a Calabi-Yau 3-fold and its deviation from Joyce's criteria: Potential image-support rigidity of A-branes that wrap around a sL $S^3$

Using a hyperKähler rotation on complex structures of a Calabi-Yau 2-fold and rolling of an isotropic 2-submanifold in a symplectic 6-manifold, we construct, by gluing, a natural family of immersed Lagrangian deformations of a branched covering of a special Lagrangian 3-sphere in a Calabi-Yau 3-fold and study how they deviate from being deformable to a family of special Lagrangian deformations by examining in detail Joyce's criteria on this family. The result suggests a potential image-support rigidity of A-branes that wrap around a special Lagrangian 3-sphere in a Calabi-Yau 3-fold, which res…