Abstract
In this paper, it is shown that the first nonzero eigenvalue λ
1
of the Laplacian operator on a compact immersed minimal hypersurface
M
in the unit sphere
S
n
+1
satisfies one of the following
where
k
0
is the infimum of the sectional curvatures of
M
. It is also shown that a compact immersed minimal hypersurface of the unit sphere
S
n
+1
with λ
1
=
n
is either isometric to the unit sphere
S
n
or else
k
0
<
n
−1
(
n
−1).