Willmore-type variational problem for foliated hypersurfaces
Fuente:
arXiv
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866914693072814080 |
|---|---|
| author | Rovenski, Vladimir |
| author_facet | Rovenski, Vladimir |
| contents | We study new Willmore-type variational problem for a hypersurface $M$ in $\mathbb{R}^{n+1}$ equipped with an $s$-dimensional foliation ${\cal F}$. Its general version is the Reilly-type functional $WF_{n,s}=\int_M F(σ^{\cal F}_1,\ldots,σ^{\cal F}_s)\,{\rm d}V$, where $σ^{\cal F}_i$ are elementary symmetric functions of the eigenvalues of the second fundamental form restricted on the leaves of $\cal F$. The first and second variations of such functionals are calculated, conformal invariance of some of $WF_{n,s}$ is also shown. The Euler-Lagrange equation for a critical hypersurface with a transversally harmonic (e.g., Riemannian) foliation $\cal F$ is found and examples with $s\le2$ and $s=n$ are considered. Critical hypersurfaces of revolution are found, and it is shown that they are a local minimum for special variations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_17565 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Willmore-type variational problem for foliated hypersurfaces Rovenski, Vladimir Differential Geometry 53C12, 53C15, 53C42 We study new Willmore-type variational problem for a hypersurface $M$ in $\mathbb{R}^{n+1}$ equipped with an $s$-dimensional foliation ${\cal F}$. Its general version is the Reilly-type functional $WF_{n,s}=\int_M F(σ^{\cal F}_1,\ldots,σ^{\cal F}_s)\,{\rm d}V$, where $σ^{\cal F}_i$ are elementary symmetric functions of the eigenvalues of the second fundamental form restricted on the leaves of $\cal F$. The first and second variations of such functionals are calculated, conformal invariance of some of $WF_{n,s}$ is also shown. The Euler-Lagrange equation for a critical hypersurface with a transversally harmonic (e.g., Riemannian) foliation $\cal F$ is found and examples with $s\le2$ and $s=n$ are considered. Critical hypersurfaces of revolution are found, and it is shown that they are a local minimum for special variations. |
| title | Willmore-type variational problem for foliated hypersurfaces |
| topic | Differential Geometry 53C12, 53C15, 53C42 |
| url | https://arxiv.org/abs/2402.17565 |