https://neckpinch.com/authors/peibiao-zhao/New arXiv papers by Peibiao Zhao, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2602.07989arxiv:2602.07989Sun, 08 Feb 2026 00:00:00 GMTLinlin Fan, Peibiao Zhao — Wang, Weng and Xia[Math. Ann. 388 (2024), no. 2] studied a mean curvature type flow for the smooth, embedded capillary hypersurfaces with a constant contact angle θ∈(0,π) and confirmed the existence of solutions by the standard PDE theory. In the present paper, we study a fractional mean curvature flow for C^(1,1)-regular hypersurfaces with a capillary-type boundary condition and obtain the short time existence by the fixed point argument.Mean curvature flowVolume-preserving & constrained flowsAnisotropic & nonlocal flowsBoundaries & non-compactFree boundary & capillaryHarmonic maps & DeTurck trick 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/2509.23865arxiv:2509.23865Sun, 28 Sep 2025 00:00:00 GMTJingshi Cui, Peibiao Zhao — In this paper, we introduce a kind of inverse mean curvature flow (1.2) in a Sasakian sub-Riemannian 3-manifold M for Legendrian curves, which slightly differs from the classical one, and confirm that this flow preserves the Legendrian condition and increases the length of curves. We establish the long-time existence of the flow (1.2) when the Webster scalar curvature W of M satisfies W ∈ (-∞, W₀ )∪ \ 0\ ∪ (W₀, +∞), where W₀ <0 and W₀ >0 are constants. Moreover, we derive that the local limit curve (the asymptotic behavior) along the flow (1.2) is a geodesic of vanishing curvature when W ≥ 0, wherea it is a geodesic of nonvanishing curvature when W is a negative constant. Specially, in the first Heisenberg group M(0), we further construct a length-preserving flow (1.3) via a dilation of the flow (1.2) and show that closed Legendrian curves converge to Euclidean helices with vertical axis. By exploiting the properties of the flow (1.3), we establish a Minkowski-type formula for Legendrian curves in M(0) and provide a new proof of the fact that the total curvature of γ⊂ M(0) with strictly positive curvature equals 2π.Mean curvature flowOther curvature flowsInverse mean curvature flowVolume-preserving & constrained flows3-manifoldsScalar curvatureLong-time existence & convergenceSasakian & contact geometry 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/2306.15469arxiv:2306.15469Tue, 27 Jun 2023 00:00:00 GMTJingshi Cui, Peibiao Zhao — Huisken and Ilmanen [J. Differential Geom., 2001] created the theory of weak solutions for inverse mean curvature flows (IMCF) of hypersurfaces on Riemannian manifolds, and proved successfully a Riemannian version of the Penrose inequality. The present paper investigates and constructs a sub-Riemannian version of the theory of weak solutions for inverse mean curvature flows of surfaces in the first Heisenberg group ℍ¹. The level set formulation of the IMCF in ℍ¹ is given by (0.1), where Ω⊂ ℍ¹ is an open set with smooth boundary, and Ω^(c) = ℍ¹∖ Ω= \ u ≤ 0\ is bounded. Let wₚ = exp ( (uₚ)/(1-p)) and wₚ satisfies (0.2). Following the argument by Moser, the key ingredient in proving the existence of weak solutions to (0.1) is to establish a uniform interior estimate for |∇₀ uₚ|. However, due to the lack of boundary continuity of |∇₀ uₚ| = (p-1)frac|∇₀ wₚ|wₚ ∈ C^(0,β)(Ω) (0< β<1, p>1) by Zhong and Mukherjee [Anal. PDE, 2021], the standard method in [R. Moser, J. Eur. Math. Soc., 2007] cannot be applied to obtain a uniform interior estimate for |∇₀ uₚ|. Fortunately, the present paper discovers two refined inequalities: Harnack inequality and Lipschitz estimate for wₚ, which allow one to obtain interior estimates for |∇₀ uₚ| independent of p. By further combining them with Arzelgraverm a-Ascoli theorem, the weak solution of (0.1) can then be generated as the limit of uₚ as p → 1, where wₚ = exp ( (uₚ)/(1-p)) and wₚ is of solutions to (0.2). As an important application of the IMCF in ℍ¹, a positive answer to an open problem posed in [F. Montefalcon, Ann. Mat. Pura Appl. (4), 2014]:Heintze-Karcher inequality in ℍ¹ is provided.Mean curvature flowOther curvature flowsInverse mean curvature flowHarnack inequalitiesWeak & singular flowsGeometric 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 https://arxiv.org/abs/2211.00855arxiv:2211.00855Wed, 02 Nov 2022 00:00:00 GMTJingshi Cui, Peibiao Zhao — In the present paper, we first establish and verify a new sharp hyperbolic version of the Michael-Simon inequality for mean curvatures in hyperbolic space ℍⁿ⁺¹ based on the locally constrained inverse curvature flow introduced by Brendle, Guan and Li, provided that M is h-convex and f is a positive smooth function, where λ^(')(r)=rmcoshr. In particular, when f is of constant, (0.1) coincides with the Minkowski type inequality stated by Brendle, Hung, and Wang. Further, we also establish and confirm a new sharp Michael-Simon inequality for the k-th mean curvatures in ℍⁿ⁺¹ by virtue of the Brendle-Guan-Li's flow, provided that M is h-convex and Ω is the domain enclosed by M. In particular, when f is of constant and k is odd, (0.2) is exactly the weighted Alexandrov-Fenchel inequalities proven by Hu, Li, and Wei.Other curvature flowsInverse mean curvature flowVolume-preserving & constrained flowsGeometric inequalities https://arxiv.org/abs/2205.12582arxiv:2205.12582Wed, 25 May 2022 00:00:00 GMTJingshi Cui, Peibiao Zhao — Brendle [6] successfully establishes the sharp Michael-Simon inequality for mean curvature on Riemannian manifolds with nonnegative sectional curvature (𝓚 ≥ 0), and the proof relies on the Alexandrov-Bakelman-Pucci method. Nevertheless, this result cannot be extended to hyperbolic space ℍⁿ⁺¹ (𝓚 = −1), as demonstrated by Counterexample 1.7. In the present paper, we propose Conjectures 1.8 and 1.9 concerning the hyperbolic version of the sharp Michael-Simon type inequality for k-th mean curvatures. However, the proof method in failed to verify the validity of these conjectures. Recently, the authors [12] proved Conjectures 1.8 and 1.9 only for h-convex hypersurfaces by means of the Brendle-Guan-Li's flow. This paper aims to utilize other types of curvature flows to prove Conjectures 1.8 and 1.9 for hypersurfaces with weaker convexity conditions. For k = 1, we first investigate a new locally constrained mean curvature flow (1.9) in ℍⁿ⁺¹ and prove its longtime existence and exponential convergence. Then, the sharp Michael-Simon type inequality for mean curvature of starshaped hypersurfaces in ℍⁿ⁺¹ is confirmed through the flow (1.9). For k ≥ 2, the sharp Michael-Simon inequality for k-th mean curvatures of starshaped, strictly k-convex hypersurfaces in ℍⁿ⁺¹ is proven using the locally constrained inverse curvature flow (1.11) introduced by Scheuer and Xia [31].Mean curvature flowOther curvature flowsInverse mean curvature flowVolume-preserving & constrained flowsLong-time existence & convergence https://arxiv.org/abs/2203.05105arxiv:2203.05105Thu, 10 Mar 2022 00:00:00 GMTBin Chen, Jingshi Cui, Peibiao Zhao — The present paper introduces a new class of geometric measures, the k-th (p,q)-mixed curvature measures, and a natural correspondence-(p,q)-Christoffel-Minkowski problem is proposed. The (p,q)-Christoffel-Minkowski problem posed here can be regarded as a natural generalization of the L_p Christoffel-Minkowski problem and Lp dual Minkowski problem. We investigate and arrive at the existence of smooth solution to the (p,q)-Christoffel-Minkowski problem by a type of expanding curvature flow. Furthermore, the uniqueness result of solutions to the (p,q)-Christoffel-Minkowski problem shall be discussed.Other curvature flowsGauss curvature flowLimit spaces & RCDStability & uniqueness https://arxiv.org/abs/2111.13281arxiv:2111.13281Fri, 26 Nov 2021 00:00:00 GMTBin Chen, Peibiao Zhao — In this paper we first obtain the existence of smooth solutions to Orlicz-Aleksandrov problem via a Gauss-like curvature flow.Other curvature flowsGauss curvature flow https://arxiv.org/abs/2111.00938arxiv:2111.00938Fri, 29 Oct 2021 00:00:00 GMTJ. Cui, P. Zhao — In this paper, we first investigate a new locally constrained mean curvature flow (1.5) and prove that if the initial hypersurface is of smoothly compact starshaped, then the solution of the flow (1.5) exists for all time and converges to a sphere in smooth topology. Following this flow argument, not only do we achieve a new proof of the celebrated sharp Michael-Simon inequality for mean curvature in (n+1) dimensional Euclidean space, but we also get the necessary and sufficient condition for the establishment of the equality. In the second part of this paper, we study a mean curvature type flow (1.7) of static convex hypersurfaces in (n+1) dimensional Euclidean space, and prove that the flow (1.7) has a unique smooth solution for all time t>0, and the static convexity of the hypersurface is preserved along the flow (1.7). Moreover, The solution of the flow (1.7) converges exponentially to a sphere of radius R in smooth topology as time tends to infinity. By exploiting the properties of this flow, we develop and present a new sharp Michael-Simon inequality for kth mean curvature.Mean curvature flowVolume-preserving & constrained flowsLong-time existence & convergence