In this paper we investigate an eigenvalue problem which appears naturally
when one considers the second variation of a constant mean curvature immers
ion. In this geometric context, the second variation operator is of the for
m Delta g + b, where b is a real valued function, and it is viewed as actin
g on smooth functions with compact support and with mean value zero. The co
ndition on the mean value comes from the fact that the variations under con
sideration preserve some balance of volume. This kind of eigenvalue problem
is interesting in itself. In the case of a compact manifold, possibly with
boundary, we compare the eigenvalues of this problem with the eigenvalues
of the usual (Dirichlet) problem and we in particular show that the two spe
ctra are interwined (in fact strictly interwined generically). As a by-prod
uct of our investigation of the case of a complete manifold with infinite v
olume we prove, under mild geometric conditions when the dimension is at le
ast 3, that the strong and weak Morse indexes of a constant mean curvature
hypersurface coincide. (C) 2000 Editions scientifiques et medicales Elsevie
r SAS.