Eigenvalue and ``twisted'' eigenvalue problems, applications to CMC surfaces (Q1583573)

Let \((M,g)\) be a Riemannian manifold with boundary and let \(b\) be a continuous real valued function on \(M\). The authors investigate an eigenvalue problem which appears naturally when one considers the second variation of a constant mean curvature immersion. In the geometric context, the second variation operator is of the form \({\Delta}_g+b\) and it is viewed as acting on smooth functions with compact support and with mean value zero. The considered eigenvalue problem is called the twisted eigenvalue problem. In the case of a compact manifold, possibly with boundary, the basic properties of the twisted Dirichlet eigenvalues and eigenfunctions are stated in Proposition 2.2, in parallel with those of the ordinary Dirichlet problem. In particular, the authors show that the two spectra are interwinted (in fact strictly interwinted generically). Finally, by using the results obtained in the case of a complete manifold with infinite volume (Section 3), the authors 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 (Theorems 4.1 and 4.2).