https://neckpinch.com/authors/tianci-luo/New arXiv papers by Tianci Luo, 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/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/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