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

TL;DR

This work extends the Morse index theory for constant mean curvature (CMC) hypersurfaces to a global setting by allowing continuous, topologically varying enlargements of domains $D(t)$ within the ambient CMC surface. It introduces a Sobolev-variational framework on generalized Lipschitz domains, including a twisted stability operator $\widetilde{L}$ and the associated bilinear form $\widetilde{I}$, and proves a Sobolev continuity theorem ensuring the $t$-continuity of the Sobolev spaces $H_t$ and the eigenvalues $\widetilde{\lambda}_k(t)$ along a $C^{0}$-monotone continuum. The core contributions are (i) The global Morse index theorem (Theorem A) linking cumulative nullities to the distribution of Jacobi fields on expanding domains; (ii) a distribution theorem for Jacobi fields (Theorem J) detailing where nontrivial Jacobi fields occur between adjacent eigenvalue thresholds and quantifying their multiplicities; and (iii) a detour approach via set-continuity that controls domain-deformation topology while preserving analytic continuities. Collectively, these results provide structural insight into Jacobi fields on variable-domain slices of a CMC hypersurface and extend Smale–Frid–Thayer-type Morse theory to a global, topologically flexible setting with explicit spectral-geometry connections.

Abstract

In this paper, we establish a "global" Morse index theorem. Given a hypersurface $M^{n}$ of constant mean curvature, immersed in $\mathbb{R}^{n+1}$. 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 $M^{n}$.

Figures (14)

• Figure 1:
• Figure 2:
• Figure 3: simple Lipschitz domain
• Figure 4: generalized Lipschitz domain (local version)
• Figure 5: (also related to Theorem S and Remark \ref{['R4.1']})

Theorems & Definitions (50)

• Definition 1.1
• Definition 1.2: Unstable cones
• Definition 1.3: Jacobi fields
• Definition 1.4: Conjugate boundaries
• Remark 1.5
• Definition 1.6
• Proposition 1.7
• proof
• Proposition 1.8
• proof