λ-超曲面的一个积分等式

A Rigidity Theorem ofλ-Hypersurfaces

研究了λ-超曲面,得到了有关完备的λ-超曲面的一个积分等式:若X:M→R^n+1是n-维完备的具有多项式面积增长的λ-超曲面且满足S有界,则有∫M(|▽H|^2+(H-λ)(H+S(λ-H)))e-|X|^2/2 dμ=0,其中,H是M的平均曲率,S是M的第二基本形式模长平方.并由该积分等式得到了一个刚性结果.

λ-hypersurfaces are studied and a rigidity result about completeλ-hypersurfaces is given.If X:M→R^n+1 is an n-dimensional completeλ-hypersurface with polynomial area growth and satisfies S bounded,then∫M(|▽H|^2+(H-λ)(H+S(λ-H)))e-|X|^2/2 dμ=0,where H is the mean curvature of M,S is the squared norm of the second fundamental form of M.As an application of the integral equation,a rigidity result about completeλ-hypersurfaces is obtained.