The Minkowski problem for the $k$-torsional rigidity
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911549262659584 |
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| author | Zhao, Xia Zhao, Peibiao |
| author_facet | Zhao, Xia Zhao, Peibiao |
| contents | P. Salani [Adv. Math., 229 (2012)] introduced the $k$-torsional rigidity associated with a $k$-Hessian equation and obtained the Brunn-Minkowski inequalities $w.r.t.$ the torsional rigidity in $\mathbb{R}^3$. Following this work, we first construct, in the present paper, a Hadamard variational formula for the $k$-torsional rigidity with $1\leq k\leq n-1$, then we can deduce a $k$-torsional measure from the Hadamard variational formula. Based on the $k$-torsional measure, we propose the Minkowski problem for the $k$-torsional rigidity and confirm the existence of its smooth non-even solutions by the method of a curvature flow. Specially, a new proof method for the uniform lower bound estimation in the $C^0$ estimation for the solution to the curvature flow is presented with the help of invariant functional $Φ(Ω_t)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_24494 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Minkowski problem for the $k$-torsional rigidity Zhao, Xia Zhao, Peibiao Differential Geometry 52A20 \ \ 35K96 \ \ 58J35 P. Salani [Adv. Math., 229 (2012)] introduced the $k$-torsional rigidity associated with a $k$-Hessian equation and obtained the Brunn-Minkowski inequalities $w.r.t.$ the torsional rigidity in $\mathbb{R}^3$. Following this work, we first construct, in the present paper, a Hadamard variational formula for the $k$-torsional rigidity with $1\leq k\leq n-1$, then we can deduce a $k$-torsional measure from the Hadamard variational formula. Based on the $k$-torsional measure, we propose the Minkowski problem for the $k$-torsional rigidity and confirm the existence of its smooth non-even solutions by the method of a curvature flow. Specially, a new proof method for the uniform lower bound estimation in the $C^0$ estimation for the solution to the curvature flow is presented with the help of invariant functional $Φ(Ω_t)$. |
| title | The Minkowski problem for the $k$-torsional rigidity |
| topic | Differential Geometry 52A20 \ \ 35K96 \ \ 58J35 |
| url | https://arxiv.org/abs/2505.24494 |