張劍鋒,洪濤清

(麗水學(xué)院數(shù)學(xué)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(xué)院數(shù)學(xué)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(zhì),通過Laplace算子的計算,得到一個關(guān)于第二基本形式模長平方S的拼擠定理,推廣了已有的結(jié)果.

局部對稱;超曲面;完備;全臍

1 引言

2 預(yù)備知識

設(shè)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(biāo)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(biāo)的變化范圍如下:

其中等號成立當(dāng)且僅當(dāng)至少n-1個ai相等.

引理2.2[6,7]設(shè)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(dāng)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(yīng)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(xué)基金(Y6100218);麗水學(xué)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