A global Morse index theorem and applications to Jacobi fields on CMC surfaces
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915212990349312 |
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| author | Huang, Wu-Hsiung |
| author_facet | Huang, Wu-Hsiung |
| contents | 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 $M^{n}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_19282 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A global Morse index theorem and applications to Jacobi fields on CMC surfaces Huang, Wu-Hsiung Differential Geometry 53C23, 58E12, 35J25, 46E35, 49R05 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 $M^{n}$. |
| title | A global Morse index theorem and applications to Jacobi fields on CMC surfaces |
| topic | Differential Geometry 53C23, 58E12, 35J25, 46E35, 49R05 |
| url | https://arxiv.org/abs/2503.19282 |