https://neckpinch.com/authors/ziyi-zhao/New arXiv papers by Ziyi Zhao, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2607.03383arxiv:2607.03383Fri, 03 Jul 2026 00:00:00 GMTPanagiota Daskalopoulos, Wenkui Du, Natasa Sesum, Ziyi Zhao — We obtain the unique asymptotics of SO(k)× SO(n-k+1)-invariant, compact, simply-connected, factorwisely non-self-similar n-dimensional κ-solutions of the Ricci flow (Mⁿ, g(t)), where n≥ 4 and 2≤ k≤ n-2. More precisely, these κ-solutions are either ancient ovals of the Ricci flow that are diffeomorphic to the standard sphere Sⁿ, having a positive curvature operator metric g(t) and a cylindrical tangent flow at -∞, or they are a Riemannian product of a Perelman's ancient oval and a shrinking round sphere. The metric g(t) of every SO(k)× SO(n-k+1)-invariant ancient oval is represented in the form g(t)=dz⊗ dz + F²(z,t) g_(S^(k-1)) + G²(z,t)g_(S^(n-k)) (up to flipping k-1 and n-k). We obtain results about the blowdown limits of such solutions, establish the unique sharp asymptotics of the profile function G(z, t), and prove that the uniqueness of G(z, t) implies the uniqueness of F(z, t). In particular, this provides the first instance of a classification result for geometric flows represented by a coupled PDE system, opening new avenues for studying the classification of higher-dimensional κ-solutions of the Ricci flow.Ricci flowAncient solutionsSingularitiesSphere theoremsStability & uniqueness https://arxiv.org/abs/2601.09441arxiv:2601.09441Wed, 14 Jan 2026 00:00:00 GMTBeomjun Choi, Wenkui Du, Ziyi Zhao — We study compact non-selfsimilar ancient noncollapsed solutions to the mean curvature flow in ℝⁿ⁺¹, called ancient ovals. Our main result is the classification of k-ovals: any k-oval (characterized by having cylindrical blow down ℝ^k× S^(n-k) and the quadratic bending asymptotics) belongs, up to space-time rigid motions and parabolic dilations, to the family of ancient ovals constructed by Haslhofer and the second author. Assuming the nonexistence of exotic ovals (recently proved by Bamler-Lai), this yields a classification of all ancient ovals and identifies the moduli space, modulo symmetries, with an open (k-1)-simplex modulo the symmetry of simplex. Although these conclusions are contained in the recent breakthrough of Bamler-Lai classifying all ancient asymptotically cylindrical flows and resolving the mean convex neighborhood conjecture, we give an alternative argument for the independently obtained classification of k-ovals in arbitrary dimensions based on a different spectral parametrization.Mean curvature flowAncient solutions(Non)collapsingEigenvalues & spectrum https://arxiv.org/abs/2404.18494arxiv:2404.18494Mon, 29 Apr 2024 00:00:00 GMTZiyi Zhao, Xiaohua Zhu — By Perelman's 𝓛-geodesic theory, we study the blow-down solutions on a noncompact κ-noncollapsed steady gradient Ricci soliton (Mⁿ, g) (n≥ 4) with nonnegative curvature operator and positive Ricci curvature away from a compact set of M. We prove that any compact split ancient solution of codimension one from the blow-down of (M, g) is of type I. The result is a generalization of our previous work from n=4 to any dimension.Ricci flowRicci solitonsAncient solutionsSingularities(Non)collapsingBoundaries & non-compact https://arxiv.org/abs/2402.00316arxiv:2402.00316Thu, 01 Feb 2024 00:00:00 GMTZiyi Zhao, Xiaohua Zhu — Let (Mⁿ,g) (n≥ 4) be a complete noncompact κ-noncollapsed steady Ricci soliton with rmRm≥ 0 and rmRic> 0 away from a compact set K of M. We prove that there is no any (n-1)-dimensional compact split limit Ricci flow of type I arising from the blow-down of (M, g), if there is an (n-1)-dimensional noncompact split limit Ricci flow. Consequently, the compact split limit ancient flows of type I and type II cannot occur simultaneously from the blow-down. As an application, we prove that (Mⁿ,g) with rmRm≥ 0 must be isometric the Bryant Ricci soliton up to scaling, if there exists a sequence of rescaled Ricci flows (M,g_(pᵢ)(t); pᵢ) of (M,g) converges subsequently to a family of shrinking quotient cylinders.Ricci flowRicci solitonsAncient solutionsSingularities(Non)collapsingBoundaries & non-compact https://arxiv.org/abs/2310.12529arxiv:2310.12529Thu, 19 Oct 2023 00:00:00 GMTZiyi Zhao, Xiaohua Zhu — In the paper, we analysis the asymptotic behavior of noncompact κ-noncollapsed steady gradient Ricci soliton (M, g) with nonnegative curvature operator away from a compact set K of M. In particular, we prove: any 4d noncompact κ-noncollapsed steady gradient Ricci soliton (M⁴, g) with nonnegative sectional curvature must be a Bryant Ricci soliton up to scaling if it admits a sequence of rescaled flows of (M⁴, g), which converges subsequently to a family of shrinking quotient cylinders.Ricci solitons(Non)collapsing4-manifoldsBoundaries & non-compact https://arxiv.org/abs/2212.02889arxiv:2212.02889Tue, 06 Dec 2022 00:00:00 GMTZiyi Zhao, Xiaohua Zhu — We introduce a new curvature-pinching condition, which is weaker than the positive sectional curvature or PIC1, and then we prove several rigidity results for the rotationally symmetric solutions of steady Ricci solitons, i.e., the Bryant Ricci solitons.Ricci solitonsCurvature pinchingSphere theoremsStability & uniquenessRotational symmetry & warped products