Faber Series for $$L^2$$ Holomorphic One-Forms on Riemann Surfaces with Boundary

Pub Date : 2024-03-22 DOI:10.1007/s40315-024-00529-4

Eric Schippers, Mohammad Shirazi

引用次数: 0

Abstract

Consider a compact surface $\mathscr {R}$ with distinguished points $z_1,\ldots ,z_n$ and conformal maps $f_k$ from the unit disk into non-overlapping quasidisks on $\mathscr {R}$ taking 0 to $z_k$. Let $\Sigma $ be the Riemann surface obtained by removing the closures of the images of $f_k$ from $\mathscr {R}$. We define forms which are meromorphic on $\mathscr {R}$ with poles only at $z_1,\ldots ,z_n$, which we call Faber–Tietz forms. These are analogous to Faber polynomials in the sphere. We show that any $L^2$ holomorphic one-form on $\Sigma $ is uniquely expressible as a series of Faber–Tietz forms. This series converges both in $L^2(\Sigma )$ and uniformly on compact subsets of $\Sigma $.

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有边界黎曼曲面上 $$L^2$$ Holomorphic One-Forms 的 Faber 系列

考虑一个紧凑曲面（$\mathscr {R}$），它有区分点（z_1,\ldots ,z_n\）和从单位盘到$\mathscr {R}$上的非重叠准星的保角映射（f_k\），取0到（z_k\）。让 $\Sigma $成为从 $\mathscr {R}$ 上移除 $f_k$ 的图像的闭包得到的黎曼曲面。我们定义了在$\mathscr {R}$上只在$z_1,\ldots ,z_n$处有极点的非定常形式，我们称之为法布尔-铁茨形式。它们类似于球面上的法布尔多项式。我们证明了任何在$\Sigma $上的$L^2$全形一元形式都可以唯一地表达为法伯-铁茨形式的数列。这个数列既在$L^2(\Sigma)$中收敛，又在$\Sigma$的紧凑子集上均匀收敛。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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