Branched covers of hyperbolic manifolds and harmonic maps

Quillen metrics and branched coverings

We give a formula to compare the Quillen metrics associated to a branched covering from holomorphic line bundles.

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 $\mathbb S^2$.

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.

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…

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。

Zoll Metrics, Branched Covers, and Holomorphic Disks

We strengthen our previous results regarding the moduli spaces of Zoll metrics and Zoll projective structures on S^2. In particular, we describe a concrete, open condition which suffices to guarantee that a totally real embedding of RP^2 in CP_2 arises from a unique Zoll projective structure on the 2-sphere. Our methods ultimately reflect the special role such structures play in the initial value problem for the 3-dimensional Lorentzian Einstein-Weyl equations.