Compact hypersurfaces in euclidean space and some inequalities

Abstract

Let (M,g) be a compact immersed hypersurface of (Rn+1,<,>), lambda(1) the first nonzero eigenvalue, alpha the mean curvature, p 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) [GRAPHICS] 2) [GRAPHICS] 3) If the scalar curvature S and the first nonzero eigenvalue lambda(1) satisfy S = lambda(1)(n - 1), then [GRAPHICS] 4) Suppose that the Ricci curvature of M is bounded below by a positive constant k. Thus [GRAPHICS] 5) Suppose that the Ricci curvature is bounded. and the scalar curvature satisfy S = A, (n - 1) and L=k2S>O is a constant. Thus [GRAPHICS]