| Autor: |
Rongli, Huang, Changzheng, Qu, Zhizhang, Wang, Weifeng, Wo |
| Rok vydání: |
2024 |
| Předmět: |
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| Druh dokumentu: |
Working Paper |
| Popis: |
In this paper, we investigate Hessian curvature hypersurfaces with prescribed Gauss images. Given geodesically strictly convex bounded domains $\Omega$ in $\mathbb{R}^n$ and $\tilde{\Omega}$ in the unit hemisphere, we prove that there is a strictly convex graphic hypersurface defined in $\Omega$ with prescribed $k$-Hessian curvatures such that its Gauss image is $\tilde{\Omega}$. 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. |
| Databáze: |
arXiv |
| Externí odkaz: |
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