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arXiv · 2505.24494

The Minkowski problem for the $k$-torsional rigidity

Xia Zhao·Peibiao Zhao
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Abstract

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 $Φ(Ω_t)$.

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Xia Zhao, Peibiao Zhao. 2026-03-28. The Minkowski problem for the $k$-torsional rigidity. https://arxiv.org/abs/2505.24494

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