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 $M^n$.