Hessian curvature hypersurfaces with prescribed Gauss image
Fuente:
arXiv
Guardado en:
| Autores principales: | , , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2024
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866910742853189632 |
|---|---|
| author | Rongli, Huang Changzheng, Qu Zhizhang, Wang Weifeng, Wo |
| author_facet | Rongli, Huang Changzheng, Qu Zhizhang, Wang Weifeng, Wo |
| contents | In this paper, we investigate Hessian curvature hypersurfaces with prescribed Gauss images. Given geodesically strictly convex bounded domains $Ω$ in $\mathbb{R}^n$ and $\tildeΩ$ in the unit hemisphere, we prove that there is a strictly convex graphic hypersurface defined in $Ω$ with prescribed $k$-Hessian curvatures such that its Gauss image is $\tildeΩ$. Our proof relies on a novel $C^2$ boundary estimate which utilizes the orthogonal invariance of hypersurfaces. Indeed, we employ some special vector fields generated by the infinitesimal rotations in $\mathbb{R}^{n+1}$ to establish the boundary $C^2$ estimates. This new approach enables us to handle the additional negative terms that arise when taking second order derivatives near the boundary. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_09159 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hessian curvature hypersurfaces with prescribed Gauss image Rongli, Huang Changzheng, Qu Zhizhang, Wang Weifeng, Wo Differential Geometry In this paper, we investigate Hessian curvature hypersurfaces with prescribed Gauss images. Given geodesically strictly convex bounded domains $Ω$ in $\mathbb{R}^n$ and $\tildeΩ$ in the unit hemisphere, we prove that there is a strictly convex graphic hypersurface defined in $Ω$ with prescribed $k$-Hessian curvatures such that its Gauss image is $\tildeΩ$. Our proof relies on a novel $C^2$ boundary estimate which utilizes the orthogonal invariance of hypersurfaces. Indeed, we employ some special vector fields generated by the infinitesimal rotations in $\mathbb{R}^{n+1}$ to establish the boundary $C^2$ estimates. This new approach enables us to handle the additional negative terms that arise when taking second order derivatives near the boundary. |
| title | Hessian curvature hypersurfaces with prescribed Gauss image |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2412.09159 |