Null curves and directed immersions of open Riemann surfaces
Null curves and directed immersions of open Riemann surfaces
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开黎曼曲面的零曲线和定向浸入
DOI：
10.1007/s00222-013-0478-8
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发表时间：
2012
影响因子：
3.1
通讯作者：
F. Forstnerič
中科院分区：
文献类型：
--
作者：
A. Alarcón;F. Forstnerič
本文研究开黎曼曲面到C^n中的全纯浸入，其导数位于C^n的一个离原点光滑的圆锥代数子簇A中。这类A-浸入的经典例子包括C^3中与R^3中极小曲面密切相关的零曲线，以及SL_2（C）中与Bryant曲面相关的零曲线。本文建立了加边黎曼曲面的A-浸入集的基本结构定理，并证明了几个逼近和去奇异化定理。假设A是不可约的，并且不包含在任何超平面中，我们证明了每个A-浸入都可以用A-嵌入来近似;这特别适用于C^3中的零曲线。如果A-{0}是Oka流形，则证明A-浸入满足Oka原理，包括Runge和Mergelyan逼近定理。另一个版本的Oka原则成立时，A承认一个光滑的Oka超平面部分。这让我们可以特别证明，每个开黎曼曲面都是双全纯的，它是C^3中一条适当嵌入的零曲线。
In this paper we study holomorphic immersions of open Riemann surfaces into C^n whose derivative lies in a conical algebraic subvariety A of C^n that is smooth away from the origin. Classical examples of such A-immersions include null curves in C^3 which are closely related to minimal surfaces in R^3, and null curves in SL_2(C) that are related to Bryant surfaces. We establish a basic structure theorem for the set of all A-immersions of a bordered Riemann surface, and we prove several approximation and desingularization theorems. Assuming that A is irreducible and is not contained in any hyperplane, we show that every A-immersion can be approximated by A-embeddings; this holds in particular for null curves in C^3. If in addition A-{0} is an Oka manifold, then A-immersions are shown to satisfy the Oka principle, including the Runge and the Mergelyan approximation theorems. Another version of the Oka principle holds when A admits a smooth Oka hyperplane section. This lets us prove in particular that every open Riemann surface is biholomorphic to a properly embedded null curve in C^3.
复杂分析多个变量
DOI：
--
发表时间：
2004
期刊：
影响因子：
--
作者：
K-H Fichtner;W.Freudenberg;M.Ohya;Ed.K.Miyajima
通讯作者：
Ed.K.Miyajima
影响因子：
4.9
作者：
M. Umehara;Kotaro Yamada
通讯作者：
M. Umehara;Kotaro Yamada