黎曼曲面上叶状结构对的全纯实现

中文摘要 AI 辅助

设 $X$ 为双曲黎曼曲面，$\u03bc$ 和 $\u03bd$ 为 $X$ 上与具有有限 Dirichlet 积分的测度叶状结构同伦的叶状结构。我们证明 $\u03bc$ 和 $\u03bd$ 是填充的当且仅当存在到另一黎曼曲面的同胚 $f:X\ o Y$ 以及 $Y$ 上可积的全纯二次微分 $q$（在自然等价意义下唯一），使得前推叶状结构分别同伦于 $q$ 的水平与垂直叶状结构。这将 Gardiner-Masur 的经典定理从闭曲面推广到任意曲面。此外，对偶 $\u211d$-树解释给出了两个 $\u211d$-树乘积中极小曲面渐近 Plateau 问题的解。我们构造了 $f:X\ o Y$ 不同伦于拟共形映射的例子，并给出了确保其拟共形的充分条件。我们推导出对局部拟共形映射、曲面间调和映射以及大映射类群的主要不等式应用。

英文摘要

Let $X$ be a hyperbolic Riemann surface and let $μ$ and $ν$ be laminations on $X$ homotopic to measured foliations with finite Dirichlet integral. We prove that $μ$ and $ν$ are filling if and only if there exists a homeomorphism to another Riemann surface $f:X\to Y$ and an integrable holomorphic quadratic differential $q$ on $Y$, unique up to the natural equivalence, such that the push-forward laminations are homotopic to the horizontal and vertical foliations of $q$ respectively. This extends a classical theorem of Gardiner-Masur from closed surfaces to arbitrary surfaces. As well, the dual $\mathbb{R}$-tree interpretation yields the solution of an asymptotic Plateau problem for minimal surfaces in a product of two $\mathbb{R}$-trees. We construct examples such that $f:X\to Y$ is not homotopic to a quasiconformal map, and we present sufficient conditions that ensure it is. We deduce applications to main inequalities for locally quasiconformal maps, harmonic maps between surfaces, and big mapping class groups.

发表机构

• University of North Carolina at Chapel Hill（北卡罗来纳大学教堂山分校）
• Graduate Center of the City University of New York（纽约市立大学研究生中心）
• Queens College of the City University of New York（纽约市立大学皇后学院）

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