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

Abstract

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