Equivalence problem for Bishop surfaces

作     者：HUANG XiaoJun 1, & YIN WanKe 2 1 Department of Mathematics, Rutgers University, New Brunswick, NJ 08902, USA 2 School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China 

作者机构：Department of Mathematics Rutgers University School of Mathematics and Statistics Wuhan University 

出 版 物：《Science China Mathematics》 (中国科学：数学（英文版）)

年 卷 期：2010年第53卷第3期

页      面：687-700页

核心收录：

学科分类：07[理学] 070104[理学-应用数学] 0701[理学-数学] 

基　　金：supported in part by US National Science Foundation (Grant No.0801056) supported in part by National Natural Science Foundation of China (Grant No.10901123) Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20090141120010) Ky and Yu-Fen Fan Fund from American Mathematical Society, and a research fund from Wuhan University(Grant No. 1082002) 

主　　题：equivalence problem Bishop surface Chern-Moser theory elliptic and hyperbolic complex tangents normal form and modular space 

摘      要：The paper has two parts. We first briefly survey recent studies on the equivalence problem for real submanifolds in a complex space under the action of biholomorphic transformations. We will mainly focus on some of the recent studies of Bishop surfaces, which, in particular, includes the work of the authors. In the second part of the paper, we apply the general theory developed by the authors to explicitly classify an algebraic family of Bishop surfaces with a vanishing Bishop invariant. More precisely, we let M be a real submanifold of C 2 defined by an equation of the form w = zz + 2Re(z s + az s+1 ) with s≥ 3 and a a complex parameter. We will prove in the second part of the paper that for s≥ 4 two such surfaces are holomorphically equivalent if and only if the parameter differs by a certain rotation. When s = 3, we show that surfaces of this type with two different real parameters are not holomorphically equivalent.