COMPACT HYPERSURFACES IN EUCLIDEAN SPACE AND SOME INEQUALITIES

Abstract

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