COMPACT HYPERSURFACES IN EUCLIDEAN SPACE AND SOME INEQUALITIES

Document Type: Regular Paper

Authors

Department of Mathematics, Fırat University, 23119 Elazig, Turkey

Abstract

Let (M,g ) be a compact immersed hypersurface of (Rn+1,<,>) , λ1 the first nonzero
eigenvalue, 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 the
above.
1) 1 2 2 2 2
M M
A dv dv
n
∫ ρ ≥ ∫ α ρ ,
2)
( )
2 2 1 2
M 1 M
dv S dv
n n
α ρ ≥ ρ
∫ − ∫ ,
3) If the scalar curvature S and the first nonzero eigenvalue λ1 satisfy S = λ1 (n −1) , then
[ ] 2 1 2 0
M
dv
n
∫ 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 M
dv k gradf dv vol M
n n
α 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 M
vol M k dv S dv
L L
≥ − λ ∫ ψ αρ − ∫ α ρ

Keywords