https://neckpinch.com/authors/xia-zhao/New arXiv papers by Xia Zhao, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2510.25435arxiv:2510.25435Wed, 29 Oct 2025 00:00:00 GMTXia Zhao, Peibiao Zhao — The study of the dual curvature measures [Y. Huang, E. Lutwak, D. Yang & G. Y. Zhang, Acta. Math. 216 (2016): 325-388], which connects the cone-volume measure and Aleksandrov's integral curvature, and has created a precedent for the theoretical research of the dual Brunn-Minkowski theory. Motivated by the foregoing groundbreaking works, the present paper introduces the p-th dual k-torsional rigidity associated with a k-Hessian equation and establishes its Hadamard variational formula with 1≤ k≤ n-1, which induces the p-th dual k-torsional measure. Further, based on the p-th dual k-torsional measure, this article, for the first time, proposes the p-th dual Minkowski problem of the k-torsional rigidity which can be equivalently converted to a nonlinear partial differential equation in smooth case: \beginalign f(x)=τ(|\nabla h|^2+h^2)^{\fracp-n2}h_Ω(x)|Du(ν^-1_Ω(x))|^k+1σ_n-k(h_ij(x)+h_Ω(x)δ_ij), \endalign where τ>0 is a constant, f is a positive smooth function defined on Sⁿ⁻¹ and σₙ₋ₖ is the (n-k)-th elementary symmetric function of the principal curvature radii. We confirm the existence of smooth non-even solution to the p-th dual Minkowski problem of the k-torsional rigidity for p<n-2 by the method of a curvature flow which converges smoothly to the solution of equation (). Specially, a novel approach for the uniform lower bound estimation in the C⁰ estimation for the solution to the curvature flow is presented with the help of invariant functional Φ(Ωₜ).Other curvature flowsGauss curvature flowStability & uniquenessLong-time existence & convergenceGeometric inequalities https://arxiv.org/abs/2505.24494arxiv:2505.24494Fri, 30 May 2025 00:00:00 GMTXia Zhao, Peibiao Zhao — 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 ℝ³. Following this work, we first construct, in the present paper, a Hadamard variational formula for the k-torsional rigidity with 1≤ k≤ 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 C⁰ estimation for the solution to the curvature flow is presented with the help of invariant functional Φ(Ωₜ).Other curvature flowsGauss curvature flowStability & uniquenessGeometric inequalities https://arxiv.org/abs/2411.00779arxiv:2411.00779Wed, 16 Oct 2024 00:00:00 GMTXia Zhao, Peibiao Zhao — The Minkowski problem for torsional rigidity (2-torsional rigidity) was firstly studied by Colesanti and Fimiani using variational method. Moreover, Hu also studied this problem by the method of curvature flows and obtained the existence of smooth even solutions. In addition, the smooth non-even solutions to the Orlicz Minkowski problem w. r. t q-torsional rigidity were given by Zhao et al. through a Gauss curvature flow. The dual curvature measure and the dual Minkowski problem were first posed and considered by Huang, Lutwak, Yang and Zhang in. The dual Minkowski problem is a very important problem, which has greatly contributed to the development of the dual Brunn-Minkowski theory and extended the other types dual Minkowski problem. To the best of our knowledge, the dual Minkowski problem w. r. t (q) torsional rigidity is still open because the dual (q) torsional measure is blank. Thus, it is a natural problem to consider the dual Minkowski problem for (q) torsional rigidity. In this paper, we introduce the p-th dual q-torsional measure and propose the p-th dual Minkowski problem for q-torsional rigidity with q>1. Then we confirm the existence of smooth even solutions for p<n (p≠ 0) to the p-th dual Minkowski problem for q-torsional rigidity by method of a Gauss curvature flow. Specially, we also obtain the smooth non-even solutions with p<0 to this problem.Other curvature flowsGauss curvature flowStability & uniquenessGeometric inequalities https://arxiv.org/abs/2406.05635arxiv:2406.05635Sun, 09 Jun 2024 00:00:00 GMTXia Zhao, Peibiao Zhao — Recently, Huang and Qin introduced the Gaussian chord measure and Lₚ-Gaussian chord measure by variational methods. Meanwhile, they posed Gaussian chord Minkowski problem for p=1 and used variational methods to obtain an origin-symmetric normalized measure solution for the Gaussian chord Minkowski problem. The smooth solution, up to now, to the Lₚ-Gaussian chord Minkowski problem is still open. Motivated by the forgoing works by Huang and Qin in, we propose in the present paper the Lₚ(p>0)-Gaussian chord Minkowski problem and log-Gaussian chord Minkowski problem, and obtain the smooth even solutions to these two types of problems by the method of a Gauss curvature flow.Other curvature flowsGauss curvature flow https://arxiv.org/abs/2404.19266arxiv:2404.19266Tue, 30 Apr 2024 00:00:00 GMTXia Zhao, Peibiao Zhao — The celebrated Minkowski problem for the torsional rigidity (2-torsional rigidity) was firstly studied by Colesanti and Fimiani using variational method. Moreover, Hu, Liu and Ma also studied the Minkowski problem {\it w.r.t.} 2-torsional rigidity by method of curvature flows and obtain the existence of smooth even solutions. Up to now, as far as we know, the study of the Minkowski problem for the q-torsional rigidity is still blank. In the present paper, we propose and investigate the Orlicz Minkowski problem for the q-torsional rigidity corresponding to the q-Laplace equation inspired by the foregoing works, and then confirm the existence of smooth non-even solutions to the Orlicz Minkowski problem for the q-torsional rigidity with q>1 by the method of a Gauss curvature flow.Other curvature flowsGauss curvature flowStability & uniqueness https://arxiv.org/abs/2308.05332arxiv:2308.05332Thu, 10 Aug 2023 00:00:00 GMTXia Zhao, Peibiao Zhao — The Lₚ chord Minkowski problem based on Chord measures and Lₚ chord measures introduced firstly by Lutwak, Xi, Yang and Zhang [38] is a very important and meaningful geometric measure problem in the Lₚ Brunn-Minkowski theory. Xi, Yang, Zhang and Zhao [45] using variational methods gave a measure solution when p > 1 and 0<p<1 in the symmetric case. Recently, Guo, Xi and Zhao [18] also obtained a measure solution for 0≤ p<1 by similar methods without the symmetric assumption. In the present paper, we investigate and confirm the orlicz chord Minkowski problem, which generalizes the Lₚ chord Minkowski problem by replacing p with a fixed continuous function φ:(0,∞)→(0,∞), and achieve the existence of smooth solutions to the orlicz chord Minkowski problem by using methods of Gauss curvature flows.Other curvature flowsGauss curvature flowLimit spaces & RCDGeometric inequalities https://arxiv.org/abs/2305.14830arxiv:2305.14830Wed, 24 May 2023 00:00:00 GMTBin Chen, Weidong Wang, Xia Zhao, Peibiao Zhao — In [Calc. Var., 57:5 (2018)], Hong-Ye-Zhang proposed the p-capacitary Orlicz-Minkowski problem and proved the existence of convex solutions to this problem by variational method for p∈(1,n). However, the smoothness and uniqueness of solutions are still open. Notice that the p-capacitary Orlicz-Minkowski problem can be converted equivalently to a Monge-Ampère type equation in smooth case: \beginalign fφ(h_K)|\nablaΨ|^p=τG \endalign for p∈(1,n) and some constant τ>0, where f is a positive function defined on the unit sphere 𝓢ⁿ⁻¹, φ is a continuous positive function defined in (0,+∞), and G is the Gauss curvature. In this paper, we confirm the existence of smooth solutions to p-capacitary Orlicz-Minkowski problem with p∈(1,n) for the first time by a class of inverse Gauss curvature flows, which converges smoothly to the solution of Equation (). Furthermore, we prove the uniqueness result for Equation () in a special case.Other curvature flowsGauss curvature flowKähler geometryStability & uniquenessLong-time existence & convergence