局部對稱流形中具有常平均曲率的完備超曲面

張劍鋒,洪濤清

(麗水學院數(shù)學系,浙江麗水 323000)

局部對稱流形中具有常平均曲率的完備超曲面

張劍鋒,洪濤清

(麗水學院數(shù)學系,浙江麗水 323000)

討論了局部對稱黎曼流形中具有常平均曲率的完備超曲面的性質,通過Laplace算子的計算,得到一個關于第二基本形式模長平方S的拼擠定理,推廣了已有的結果.

局部對稱;超曲面;完備;全臍

1 引言

2 預備知識

設M是等距浸入局部對稱流形Nn+1中的一個完備超曲面.在Nn+1上選擇局部單位正交標架場{e1,…,en,en+1},使得限制在M時,e1,…,en與M相切,en+1是M的法向量.約定指標的變化范圍如下:

其中等號成立當且僅當至少n-1個ai相等.

引理2.2[6,7]設M是Ricci曲率有下界的完備黎曼流形,F是M上有上界的C2-函數(shù),則對任意ε＞0,存在一點x∈M,使得

3 定理的證明

當ε趨向于0時,由于F有界,不等式右邊也趨向于0.選取點列{εm},使得εm→0(m→∞).則對應的存在點列{xm},使得點列{F(xm)}收斂(如果有必要,可以取子點列),不妨記收斂極限為F0.根據(jù)上確界的定義可知F0=supF,再根據(jù)F的定義可知

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Complete hypersurfaces with constant mean curvature in locally symmetric manifold

Zhang Jianfeng,Hong Taoqing
(Department of Mathematics,LishuiUniversity,Lishui323000,China)

In this paper,the complete hypersurfaces with constant mean curvature in locally symmetric manifold are investigated,a pinching theorem about the square of the length of the second fundamental form S is obtained by using the Laplace operator,which generalizes the known results.

locally symmetric,hypersurfaces,complete,totally umbilical

O186

A

1008-5513(2013)02-0118-07

10.3969/j.issn.1008-5513.2013.02.002

2012-11-12.

浙江省自然科學基金(Y6100218);麗水學院科研項目(KZ201113,KY201105).

張劍鋒(1972-),博士,副教授,研究方向：微分幾何.

2010 MSC:53B20,53A10