compact hypersurfaces in euclidean space and some inequalities

let (m,g ) be a compact immersed hypersurface of (rn+1,) , λ1 the first nonzeroeigenvalue, α the mean curvature, ρ the support function, a the shape operator, vol (m ) the volume of m,and s the scalar curvature of m. in this paper, we established some eigenvalue inequalities and proved theabove.1) 1 2 2 2 2m ma dv dvn∫ ρ ≥ ∫ α ρ ,2)( )2 2 1 2m 1 mdv s dvn nα ρ ≥ ρ∫ − ∫ ,3) if the scalar curvature s and the first nonzero eigenvalue λ1 satisfy s = λ1 (n −1) , then[ ] 2 1 2 0mdvn∫ α − λ ρ ≥ ,4) suppose that the ricci curvature of m is bounded below by a positive constant k. thus( )2 2 2 ( )m 1 mdv k gradf dv vol mn nα ρ ≥ +∫ − ∫ ,5) suppose that the ricci curvature is bounded and the scalar curvature satisfy s = λ1 (n −1) and l=k-2s>0 is a constant. thus( ) 1 2 2 2 2 .m mvol m k dv s dvl l≥ − λ ∫ ψ αρ − ∫ α ρ