Singularities and Topological Change for Deforming Domains in Manifolds
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912292289904640 |
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| author | Huang, Wu-Hsiung |
| author_facet | Huang, Wu-Hsiung |
| contents | Given a $C^{0}$-deformation of domains $D(t)$ on a manifold $M^{n}$, which allows the topological types of the domains $D(t)$ to change with $t$, in what cases are the entities in analysis continuous in $t$, so that analysis techniques still work along $t$? This type of problem was addressed in our previous work [Hw] concerning domains on hypersurfaces of constant mean curvature (CMC) in $\mathbb{R}^{n+1}$. In this paper, we consider a more popular circumstance, where the deforming domains are situated in any smooth manifold $M^{n}$ equipped with an arbitrary self-adjoint strongly elliptic operator $L$ (replacing the stability operator for CMC hypersurfaces in $\mathbb R^{n+1}$ [Hw]).
We define the concept of quasi-Lipschitz domains by gluing together some boundary points of a Lipschitz domain in a specific manner, allowing the topology of the deforming domain $D(t)$ to change. It is established that any ``appropriate" monotone $C^{0}$-deformation on $M^{n}$ (see Definition 1.1) exhibits Sobolev continuity and eigenvalue continuity of $L$ along $t$. As a consequence, a ``global" Morse index theorem is obtained. Furthermore, given an \emph{arbitrary} Lipschitz domain $D$ in $M^{n}$, we can find a $C^0$-deformation from a small $n$-ball to the domain $D$, such that the topology of $D(t)$ may change, yet the required continuity theorems still hold, and hence the global Morse index theorem still follows. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_17961 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Singularities and Topological Change for Deforming Domains in Manifolds Huang, Wu-Hsiung Differential Geometry 58K99, 58K60, 53C23, 35J15 Given a $C^{0}$-deformation of domains $D(t)$ on a manifold $M^{n}$, which allows the topological types of the domains $D(t)$ to change with $t$, in what cases are the entities in analysis continuous in $t$, so that analysis techniques still work along $t$? This type of problem was addressed in our previous work [Hw] concerning domains on hypersurfaces of constant mean curvature (CMC) in $\mathbb{R}^{n+1}$. In this paper, we consider a more popular circumstance, where the deforming domains are situated in any smooth manifold $M^{n}$ equipped with an arbitrary self-adjoint strongly elliptic operator $L$ (replacing the stability operator for CMC hypersurfaces in $\mathbb R^{n+1}$ [Hw]). We define the concept of quasi-Lipschitz domains by gluing together some boundary points of a Lipschitz domain in a specific manner, allowing the topology of the deforming domain $D(t)$ to change. It is established that any ``appropriate" monotone $C^{0}$-deformation on $M^{n}$ (see Definition 1.1) exhibits Sobolev continuity and eigenvalue continuity of $L$ along $t$. As a consequence, a ``global" Morse index theorem is obtained. Furthermore, given an \emph{arbitrary} Lipschitz domain $D$ in $M^{n}$, we can find a $C^0$-deformation from a small $n$-ball to the domain $D$, such that the topology of $D(t)$ may change, yet the required continuity theorems still hold, and hence the global Morse index theorem still follows. |
| title | Singularities and Topological Change for Deforming Domains in Manifolds |
| topic | Differential Geometry 58K99, 58K60, 53C23, 35J15 |
| url | https://arxiv.org/abs/2503.17961 |