Estimates For The First Eigenvalues Of Jacobi Operators On Space-like Hypersurfaces In De Sitter Spaces

In this article,we mainly study estimation problems for the first eigenvalues of two Jacobi operators JW,Jr on compact space-like hypersurface Mn in de Sitter space S1n+1(1).After reviewing some basic concepts and formulas of space-like hy-persurface in de Sitter space,we discuss respectively the first eigenvalue problems for Jacobi operators of the linear Weingarten space-like hypersurface,and the space-like hypersurface satisfying the condition(n-r)CnHr+1 + anH=b in de Sitter space,where Hr is the r-order mean curvature of Mn,Cnr is the combinatorial number,the main results are described as follows:1.We investigate the Jacobi operator Jw of the linear Weingarten space-like hypersurface in de Sitter space,and obtain the estimation of upper bound for the first eigenvalue ?1Jw of Jw.In addition,we obtain that ?1Jw can be represented by the mean curvature H,the dimension n and the constants a,b when Mn is a totally umbilical and non-totally geodesic hypersurface satisfying(n-1)H2 + aH = b.see theorems 2.1,2.2.2.We study the Jacobi operator JT of the space-like hypersurface satisfying(n-r)CnrHr+1 + anH=b(where Cnr is the combinatorial number)in de Sitter space.When Mn is totally umbilical and non-totally geodesic,we get the relationship between the first eigenvalue ?1Jr of the operator Jr and the mean curvature H,see theorem 3.1.