COMPACT HYPERSURFACES IN EUCLIDEAN SPACE AND SOME INEQUALITIES

Let (M,g ) be a compact immersed hypersurface of (Rn+1, < , > ) , ?1 the first nonzero eigenvalue, ? the mean curvature, ? 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 the above. 1) 1 2 2 2 2 M M A dv dv n ? ? ? ? ? ? , 2) ( ) 2 2 1 2 M 1 M dv S dv n n ? ? ? ? ? ? ? , 3) If the scalar curvature S and the first nonzero eigenvalue ?1 satisfy S = ?1 (n ?1) , then [ ] 2 1 2 0 M dv n ? ? ? ? ? ? , 4) Suppose that the Ricci curvature of M is bounded below by a positive constant k. Thus ( ) 2 2 2 ( ) M 1 M dv k gradf dv vol M n n ? ? ? + ? ? ? , 5) Suppose that the Ricci curvature is bounded and the scalar curvature satisfy S = ?1 (n ?1) and L=k- 2S > 0 is a constant. Thus ( ) 1 2 2 2 2 . M M vol M k dv S dv L L ? ? ? ? ? ?? ? ? ? ?