Some Recent Developments on Hopf’s Holomorphic Form

被引:0
|
作者
Manfredo P. do Carmo
机构
[1] Instituto de Matemtica Pura e Aplicada (IMPA),
来源
Results in Mathematics | 2011年 / 60卷
关键词
53A10; 53C42; Constant mean curvature; Hopf’s quadratic form; Riemann surfaces; Hopf-type theorems;
D O I
暂无
中图分类号
学科分类号
摘要
Hopf’s theorem on surfaces in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}^3}$$\end{document} with constant mean curvature (Hopf in Math Nach 4:232–249, 1950-51) was a turning point in the study of such surfaces. In recent years, Hopf-type theorems appeared in various ambient spaces, (Abresch and Rosenberg in Acta Math 193:141–174, 2004 and Abresch and Rosenberg in Mat Contemp Sociedade Bras Mat 28:283-298, 2005). The simplest case is the study of surfaces with parallel mean curvature vector in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${M_k^n \times \mathbb{R}, n \ge 2}$$\end{document} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${M_k^n}$$\end{document} is a complete, simply-connected Riemannian manifold with constant sectional curvature k ≠ 0. The case n = 2 was solved in Abresch and Rosenberg 2004. Here we describe some new results for arbitrary n.
引用
收藏
页码:175 / 183
页数:8
相关论文
共 50 条