Stability and eigenvalue estimates of linear Weingarten hypersurfaces in a sphere

Let MM be an nn-dimensional compact hypersurface without boundary in a unit sphere Sn+1(1)Sn+1(1). MM is called a linear Weingarten hypersurface if cR+dH+e=0cR+dH+e=0, where c,dc,d and ee are constants with c2+d2>0c2+d2>0, RR and HH denote the scalar curvature and the mean curvature of MM, respectively. By the Gauss equation, we can rewrite the condition cR+dH+e=0cR+dH+e=0 as (n−1)ẽH2+aH=b, where H2H2 is the 2nd mean curvature, aa, bb and ẽ are constants such that a2+ẽ2>0, when ẽ=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)H2+aH=b(n−1)H2+aH=b, where aa and bb are constants, can be characterized as critical points of the functional ∫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.