A global Morse index theorem and applications to Jacobi fields on CMC surfaces

In this paper, we establish a “global” Morse index theorem. Given a hypersurface M n of constant mean curvature, immersed in ℝ n + 1 . Consider a continuous deformation of “generalized” Lipschitz domain D ( t ) enlarging in M n . The topological type of D ( t ) is permitted to change along t , so that D ( t ) has an arbitrary shape which can “reach afar” in M n , i.e. cover any preassigned area. The proof of the global Morse index theorem is reduced to the continuity in t of the Sobolev space H t of variation functions on D ( t ) , as well as the continuity of eigenvalues of the stability operator. We devise a “detour” strategy by introducing a notion of “set-continuity” of D ( t ) in t to yield the required continuities of H t and of eigenvalues. The global Morse index theorem thus follows and provides a structural theorem of the existence of Jacobi fields on domains in M n .