带指定Scherk渐近性的从带孔黎曼曲面到双曲平面的调和映射
Harmonic Maps from Punctured Riemann Surfaces to the Hyperbolic Plane with Prescribed Scherk Asymptotics
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中文总结 AI 辅助
本文构造了从带孔黎曼曲面到双曲平面的调和映射，在孔洞处指定Scherk渐近性，并涵盖水平悬链面的调和映射。
中文摘要 AI 辅助
设$X$为亏格$g\geq 0$的紧黎曼曲面，其允许一个具有非空不动点集的反全纯对合$\iota$，并设$D = \{p_1,\dots,p_k\}\subset \text{Fix}(\iota)$。在每个孔洞处，我们指定一个与给定的偶数次且首项系数为负的实多项式二次微分相关联的Scherk映射。此处，Scherk映射是从$\mathbb{C}$到$\mathbb{H}^2$中理想多边形内部的调和微分同胚，其Hopf微分即为给定的多项式。我们构造一个调和映射$h:X\backslash D \to \mathbb{H}^2$，使得其在每个孔洞处的渐近行为与指定的Scherk映射相匹配。特别地，$h$的像在每个末端趋于一个理想多边形。所得的$h$涵盖了通过取$\mathbb{H}^2 \times \mathbb{R}$中水平悬链面的$\mathbb{H}^2$因子而获得的调和映射。
英文摘要
Let $X$ be a genus $g\geq 0$ compact Riemann surface that admits an antiholomorphic involution $ι$ with non empty fixed-point set, and let $D = \{p_1,\dots,p_k\}\subset \text{Fix}(ι)$. At each puncture, we prescribe a Scherk map associated to a given real polynomial quadratic differential with even degree and negative leading coefficient. Here a Scherk map is the harmonic diffeomorphism from $\mathbb{C}$ to the interior of an ideal polygon in $\mathbb{H}^2$, whose Hopf differential is the given polynomial. We construct a harmonic map \[ h:X\backslash D \to \mathbb{H}^2 \] whose asymptotic behavior at each puncture matches the prescribed Scherk map. In particular, the image of $h$ tends to an ideal polygon near each end. The resulting $h$ covers harmonic maps obtained by taking the $\mathbb{H}^2$ factors of horizontal catenoids in $\mathbb{H}^2 \times \mathbb{R}$.
发表机构
- Nankai University（南开大学）
机构由 AI 辅助整理，请以论文原文为准。
