Eigenvalue and "twisted" eigenvalue problems, applications to CMC surfaces

In this paper we investigate an eigenvalue problem which appears naturally when one considers the second variation of a constant mean curvature immersion. In this geometric context, the second variation operator is of the form Delta g + b, where b is a real valued function, and it is viewed as acting on smooth functions with compact support and with mean value zero. The condition on the mean value comes from the fact that the variations under consideration 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 spectra are interwined (in fact strictly interwined generically). As a by-product of our investigation of the case of a complete manifold with infinite volume we prove, under mild geometric conditions when the dimension is at least 3, that the strong and weak Morse indexes of a constant mean curvature hypersurface coincide. (C) 2000 Editions scientifiques et medicales Elsevier SAS.