Minimal surfaces and harmonic diffeomorphisms from the complex plane onto a Hadamard surface

Geometric and Martin boundaries of a Cartan-Hadamard surface

We give a general criterion for the Dirichlet problem at infinity (DPI) on a Cartan-Hadamard surface to be solvable, which we primarily use to give the best possible upper radial radial curvature bound for solvability of the DPI, but which is also flexible enough to accommodate flats. In particular, any (upper) radial curvature bound which implies transience also implies solvability of the DPI, which is perhaps surprising. Taking advantage of the structure provided by uniformization, we show that solvability of the DPI implies there is a natural continuous surjection of the Martin boundary ont…

球面锥面之间极小拉格朗日微分同胚的刚性

Rigidity of minimal Lagrangian diffeomorphisms between spherical cone surfaces

本文证明具有锥奇点的闭球面之间的极小拉格朗日微分同胚必为等距映射，并将Liebmann刚性定理推广到常正高斯曲率闭曲面的分支浸入情形。

Energy-minimal diffeomorphisms between doubly connected Riemann surfaces

Let $N=(Ω,σ)$ and $M=(Ω^*,ρ)$ be doubly connected Riemann surfaces and assume that $ρ$ is a smooth metric with bounded Gauss curvature $\mathcal{K}$ and finite area. The paper establishes the existence of homeomorphisms between $Ω$ and $Ω^*$ that minimize the Dirichlet energy. In the class of all homeomorphisms $f \colon Ω\onto Ω^\ast$ between doubly connected domains such that $\Mod Ω\le \Mod Ω^\ast$ there exists, unique up to conformal authomorphisms of $Ω$, an energy-minimal diffeomorphism which is a harmonic diffeomorphism. The results improve and extend some recent results of Iwaniec, Koh…

Wasserstein空间上的随机平行运输与在闭黎曼流形上的微分同胚群上的等变扩散

Stochastic parallel transport on the Wasserstein space and equivariant diffusions on the group of diffeomorphisms over a closed Riemannian manifold

本文研究了Wasserstein空间上的随机平行运输与微分同胚群上的等变扩散，证明了其存在性和唯一性，并展示了其分解结构。

Tree-like curves and their inflection points

We give a criterion when a planar tree-like curve, i.e. a generic immersed plane curve each double point of which cuts it into two disjoint parts, can be send by a diffeomorphism of the plane onto a curve with no inflection points. We also present some upper and lower bounds for the minimal number of inflection points on such curves unremovable by diffeomorphisms of the plane in terms of their combinatorics.

A factorization theorem for harmonic maps

Let $f$ be a harmonic map from a Riemann surface to a Riemannian $n$-manifold. We prove that if there is a holomorphic diffeomorphism $h$ between open subsets of the surface such that $f\circ h = f$, then $f$ factors through a holomorphic map onto another Riemann surface. If such $h$ is anti-holomorphic, we obtain an analogous statement. For minimal maps, this result is well known and is a consequence of the theory of branched immersions of surfaces due to Gulliver-Osserman-Royden. Our proof relies on various geometric properties of the Hopf differential.

Lipschitz regularity of energy-minimal mappings between doubly connected Riemann surfaces

Let $M$ and $N$ be doubly connected Riemann surfaces with $\mathscr{C}^{1,α}$ boundaries and with nonvanishing conformal metrics $σ$ and $\wp$ respectively, and assume that $\wp$ is a smooth metric with bounded Gauss curvature $\mathcal{K}$ and finite area. Assume that ${\Ho}_ρ(M, N)$ is the class of all $\mathscr{W}^{1,2}$ bomeomorphisms between $M$ and $N$ and assume that $\mathcal{E}^\wp: \overline{\Ho}_ρ(M, N)\to \mathbf{R}$ is the Dirichlet-energy functional, where $\overline{\Ho}_ρ(M, N)$ is the closure of ${\Ho}_ρ(M, N)$ in $\mathscr{W}^{1,2}(M,N)$. By using a result of Iwaniec, Kovalev…

Pitot四边上的极小曲面

Minimal surfaces over the Pitot quadrilaterals

本文提出了一种构造Pitot四边上Scherk型极小图的方法，通过调和映射和Weierstrass数据，得到具有特定高斯映射和高度函数行为的极小图，并证明了其曲率最大化性质。