https://neckpinch.com/authors/rong-zhou/New arXiv papers by Rong Zhou, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2610.35707arxiv:2610.35707Mon, 26 Oct 2026 00:00:00 GMTTianci Luo, Yong Wei, Ruijia Zhang, Rong Zhou — We study quermassintegral-preserving curvature flows in ℍⁿ⁺¹, n≥3, driven by the curvature quotient (Eₖ/E_ℓ)^(1/(k-ℓ)). For 1≤ℓ<k with k+ℓ≤ n, we prove that positive sectional curvature is preserved. Starting from a smooth closed strictly convex hypersurface with positive sectional curvature, the flow exists for all time and converges smoothly and exponentially to a geodesic sphere. As an application, we establish sharp Alexandrov–Fenchel inequalities between Wₖ and W_ℓ for smooth bounded convex domains whose boundaries have nonnegative sectional curvature, whenever 0≤ℓ<k≤ n and k+ℓ≤ n. Equality holds if and only if the domain is a geodesic ball. Together with previously known cases, these inequalities give the full family of Alexandrov–Fenchel inequalities for this class of domains in ℍ⁶. We also construct counterexamples showing that k+ℓ≤ n is the optimal range for preservation of positive sectional curvature by these flows.Other curvature flowsSphere theoremsLong-time existence & convergenceGeometric inequalities https://arxiv.org/abs/2609.13829arxiv:2609.13829Sat, 12 Sep 2026 00:00:00 GMTTianci Luo, Yong Wei, Rong Zhou — We prove the Alexandrov–Fenchel inequalities between any two spherical quermassintegrals for smooth weakly convex domains contained in an open hemisphere, with equality if and only if the domain is a geodesic ball. For strictly convex hypersurfaces, we introduce a globally constrained curvature flow which preserves one quermassintegral and decreases the next one. We establish uniform curvature estimates by combining a pinching estimate, a support function argument, and spherical polarity. As a consequence, the flow exists for all time and converges smoothly and exponentially to a geodesic sphere. The monotonicity of the quermassintegrals gives the full family of Alexandrov–Fenchel inequalities. A short-time mean curvature flow approximation and a localized rigidity argument extend the result, including the equality characterization, to weakly convex domains.Mean curvature flowOther curvature flowsVolume-preserving & constrained flowsCurvature pinchingStability & uniquenessLong-time existence & convergenceGeometric inequalities https://arxiv.org/abs/2608.25971arxiv:2608.25971Wed, 26 Aug 2026 00:00:00 GMTTianci Luo, Yong Wei, Rong Zhou — Let n≥3 and 0≤τ≤2. We prove that every smooth closed connected immersed hypersurface in hyperbolic space whose induced metric has positive τ-bi-Ricci curvature admits a mean curvature flow with surgery which has only finitely many surgery times and terminates. In the range compatible with cylindrical necks, the key ingredients are a preserved quantitative spectral pinching condition, which converts the intrinsic hypothesis into uniform two-convexity, and cylindrical and derivative estimates that remain valid across the hyperbolic standard neck replacement. At the endpoint n=3, τ=2, positive τ-bi-Ricci curvature is positive Ricci curvature and forces strict convexity, so the ordinary mean curvature flow converges to a round point. Consequently, the underlying manifold is diffeomorphic to a sphere or to a finite connected sum of copies of Sⁿ⁻¹×S¹. If the initial hypersurface is embedded and bounds a compact domain, that domain is a one-handlebody, namely a ball with finitely many one-handles attached.Mean curvature flowSurgeryCurvature pinchingEigenvalues & spectrum https://arxiv.org/abs/2608.11841arxiv:2608.11841Wed, 12 Aug 2026 00:00:00 GMTChaoqun Gao, Yong Wei, Rong Zhou — We prove sharp Weinstock inequalities for the first nonzero Steklov eigenvalue of smooth outward-minimizing domains in Euclidean space and hyperbolic space. The method is based on the weak inverse mean curvature flow of Huisken–Ilmanen. The main new ingredient is an endpoint distributional monotonicity argument obtained from the calibrated weak formulation and the Gauss–Green formula for divergence-measure fields.Mean curvature flowOther curvature flowsInverse mean curvature flowEigenvalues & spectrum https://arxiv.org/abs/2607.26923arxiv:2607.26923Wed, 29 Jul 2026 00:00:00 GMTTianci Luo, Yong Wei, Rong Zhou — We study smooth compact strictly convex hypersurfaces in the unit ball that meet the support sphere orthogonally and evolve by the α-Gauss curvature flow ∂ₜX=−K^αν, α>0. We prove that the solution remains strictly convex, becomes extinct in finite time and contracts to a single point on the support sphere. If α>1/(n+2), we apply a Cayley-type conformal map that sends the extinction point to the origin of a Euclidean half-space and then normalize the enclosed half-space volume. The resulting normalized hypersurfaces converge smoothly to the unit hemisphere. The proof combines boundary identities for the spherical free boundary, a boundary-adapted Tso estimate, an almost-monotonicity formula for a half-space entropy, and uniform curvature estimates for the normalized flow.Other curvature flowsGauss curvature flowSingularitiesFree boundary & capillaryLong-time existence & convergence https://arxiv.org/abs/2607.02067arxiv:2607.02067Thu, 02 Jul 2026 00:00:00 GMTChaoqun Gao, Yong Wei, Rong Zhou — We prove an anisotropic analogue of the higher regularity theorem of Huisken and Ilmanen for inverse mean curvature flow. For an arbitrary smooth Minkowski norm, we first prove a Huisken–Ilmanen type Harnack estimate for smooth closed strictly star-shaped solutions. We then construct global smooth solutions starting from C¹ strictly star-shaped hypersurfaces with bounded nonnegative weak anisotropic mean curvature. Combining this construction with the asymptotic theory for weak inverse anisotropic mean curvature flow, we show that weak solutions starting from bounded smooth initial sets become smooth outside a compact set.Mean curvature flowOther curvature flowsInverse mean curvature flowAnisotropic & nonlocal flowsHarnack inequalitiesWeak & singular flows https://arxiv.org/abs/2604.25513arxiv:2604.25513Tue, 28 Apr 2026 00:00:00 GMTTianci Luo, Yong Wei, Rong Zhou — We study contracting curvature flows of compact hypersurfaces with positive sectional curvature in hyperbolic space ℍⁿ⁺¹. The speed is assumed to be homogeneous of degree one in the principal curvatures and to satisfy certain conditions. This class of flows includes the kth mean curvature flow as a special case. We show that if the initial hypersurface has positive sectional curvature, then this property is preserved along the flow, and the evolving hypersurface contracts to a round point in finite time.Mean curvature flowOther curvature flowsSingularitiesSphere theoremsHomogeneous spaces https://arxiv.org/abs/2510.08168arxiv:2510.08168Thu, 09 Oct 2025 00:00:00 GMTChaoqun Gao, Yong Wei, Rong Zhou — We first establish a local gradient estimate for anisotropic p-harmonic functions. A key feature of our estimate is that the constant remains bounded as p→ 1; consequently, in the limit p→ 1, this estimate yields the local gradient estimate for weak solutions of the inverse anisotropic mean curvature flow (IAMCF). As an application, we show that the weak IAMCF is asymptotic to the expanding Wulff shape solution at the infinity, thereby extending the result of Huisken and Ilmanen in [8] to the anisotropic case.Mean curvature flowAnisotropic & nonlocal flowsWeak & singular flows