Stability and eigenvalue estimates of linear Weingarten hypersurfaces in a sphere

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:397

Article

Chen, Hang³  Wang, Xianfeng^1,2 

[1] Nankai Univ, Sch Math Sci, Tianjin 300071, Peoples R China

[2] Nankai Univ, LPMC, Tianjin 300071, Peoples R China

[3] Tsinghua Univ, Dept Math Sci, Beijing 100084, Peoples R China

关键词: Linear Weingarten hypersurfaces;    Mean curvature;    Scalar curvature;    Stability;    Eigenvalue estimates;   

DOI  :  10.1016/j.jmaa.2012.08.003

来源: Elsevier

【 摘 要 】

Let M be an n-dimensional compact hypersurface without boundary in a unit sphere Sn+1(1). M is called a linear Weingarten hypersurface if cR + dH + e = 0, where c, d and e are constants with c(2)+d(2) > 0, R and H denote the scalar curvature and the mean curvature of M, respectively. By the Gauss equation, we can rewrite the condition cR + dH + e = 0 as (n - 1)(e) over tildeH(2) + aH = b, where H-2 is the 2nd mean curvature, a, b and (e) over tilde are constants such that a(2) + (e) over tilde (2) > 0, when (e) over tilde = 0, it reduces to the constant mean curvature case. In this paper, we obtain some stability results about linear Weingarten hypersurfaces, which generalize the stability results about the hypersurfaces with constant mean curvature or with constant scalar curvature. We show that linear Weingarten hypersurfaces satisfying (n - 1)H-2 + aH = b, where a and b are constants, can be characterized as critical points of the functional integral(M)(a + nH) dv for volume-preserving variations. We prove that such a linear Weingarten hypersurface is stable if and only if it is totally umbilical and non-totally geodesic. We also obtain optimal upper bounds for the first and second eigenvalues of the Jacobi operator of linear Weingarten hypersurfaces. (C) 2012 Elsevier Inc. All rights reserved.

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