Jingshi Cui — Neckpinch

Legendrian curve flow in Sasakian sub-Riemannian 3-manifolds

Sun, 28 Sep 2025 00:00:00 GMT

Jingshi 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.

Horizontal inverse mean curvature flow in the Heisenberg group

Tue, 27 Jun 2023 00:00:00 GMT

Jingshi 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.

Michael-Simon type inequalities in hyperbolic space ℍⁿ⁺¹ via Brendle-Guan-Li's flows

Wed, 02 Nov 2022 00:00:00 GMT

Jingshi 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.

Locally constrained flows and sharp Michael-Simon inequalities in hyperbolic space

Wed, 25 May 2022 00:00:00 GMT

Jingshi 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].

An expanding curvature flow and the (p,q)-Christoffel-Minkowski problems

Thu, 10 Mar 2022 00:00:00 GMT

Bin 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.

Mean curvature type flow and sharp Micheal-Simon inequalities

Fri, 29 Oct 2021 00:00:00 GMT

J. 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.