https://neckpinch.com/authors/xiaohua-zhu/New arXiv papers by Xiaohua Zhu, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2610.05201arxiv:2610.05201Sun, 04 Oct 2026 00:00:00 GMTLiding Huang, Xiaohua Zhu — We first define an extended Mabuchi functional for a class of generalized Kähler metrics of symplectic type on a toric Fano manifold (M,J,T), and prove it is proper. Then by showing the monotonicity of functional along the normalized generalized Kähler-Ricci flow for any initial T-invariant generalized Kähler metric with its symplectic form in 2πc₁(M,J), we prove that the flow converges to a Kähler-Ricci soliton. The result confirms a conjecture of Apostolov-Streets-Ustinovskiy.Ricci flowKähler–Ricci flowRicci solitonsKähler–Ricci solitonsKähler geometryKähler–Einstein & K-stability https://arxiv.org/abs/2604.17367arxiv:2604.17367Sun, 19 Apr 2026 00:00:00 GMTJintao Ye, Xiaohua Zhu — We establish a Petersen-Wei type relative volume comparison theorem for weighted Riemannian manifolds under both L^p-bounds of the Bakry-Émery Ricci curvature and the gradient of potential function. As an application, we give a modified proof for a volume comparison and monotonicity of Kähler-Ricci flow established in a recent work of Tian-Zhang-Zhang-Zhu-Zhu with improved estimate for error term.Ricci flowKähler–Ricci flowKähler geometryLimit spaces & RCD https://arxiv.org/abs/2509.14820arxiv:2509.14820Thu, 18 Sep 2025 00:00:00 GMTGang Tian, Qi S. Zhang, Zhenlei Zhang, Meng Zhu, Xiaohua Zhu — We first prove a uniform integral Laplace comparison result for the Kähler Ricci flow on Fano manifolds which depends only on the initial metric. As an application, using Cheeger-Colding theory and previous results by some of the authors, we give a direct and independent proof of the Hamilton-Tian conjecture on convergence of Kähler-Ricci flows, modulo a codimension 4 singular set. We also expounded on some existing literature on this conjecture.Ricci flowKähler–Ricci flow4-manifoldsKähler geometry 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/2305.05366arxiv:2305.05366Tue, 09 May 2023 00:00:00 GMTFeng Wang, Xiaohua Zhu — We prove that the Gromov-Hausdorff limit of Kähler-Ricci flow on a mathbf G-spherical Fano manifold X is a mathbf G-spherical ℚ-Fano variety X_(∞), which admits a (singular) Kähler-Ricci soliton. Moreover, the mathbf G-spherical variety structure of X_(∞) can be constructed as a center of torus ℂ^*-degeneration of X induced by an element in the Lie algebra of Cartan torus of mathbf G.Ricci flowKähler–Ricci flowRicci solitonsKähler–Ricci solitonsKähler geometryLimit spaces & RCD 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 https://arxiv.org/abs/2209.05029arxiv:2209.05029Mon, 12 Sep 2022 00:00:00 GMTGang Tian, Xiaohua Zhu — In this paper, we prove that on a Fano mathbf G-manifold (M,J), the Gromov-Hausdorff limit of Kähler-Ricci flow with initial metric in 2πc₁(M) must be a ℚ-Fano horosymmetric variety M_∞, which admits a singular Kähler-Ricci soliton. Moreover, M_∞ is a limit of ℂ^*-degeneration of M induced by an element in the Lie algebra of Cartan torus of mathbf G. A similar result can be also proved for Kähler-Ricci flows on any Fano horosymmetric manifolds. As an application, we generalize our previous result about the type II singularity of Kähler-Ricci flows on Fano mathbf G-manifolds to Fano horosymmetric manifolds.Ricci flowKähler–Ricci flowRicci solitonsKähler–Ricci solitonsSingularitiesKähler geometryLimit spaces & RCD https://arxiv.org/abs/2107.12680arxiv:2107.12680Tue, 27 Jul 2021 00:00:00 GMTGang Tian, Liang Zhang, Xiaohua Zhu — Let (M,J₀) be a Fano manifold which admits a Kähler-Ricci soliton, we analyze the behavior of the Kähler-Ricci flow near this soliton as we deform the complex structure J₀. First, we will establish an inequality of Lojasiewicz's type for Perelman's entropy along the Kähler-Ricci flow. Then we prove the convergence of Kähler-Ricci flow when the complex structure associated to the initial value lies in the kernel Z or negative part of the second variation operator of Perelman's entropy. As applications, we solve the Yau-Tian-Donaldson conjecture for the existence of Kähler-Ricci solitons in the moduli space of complex structures near J₀, and we show that the kernel Z corresponds to the local moduli space of Fano manifolds which are modified K-semistable. We also prove an uniqueness theorem for Kähler-Ricci solitons.Ricci flowKähler–Ricci flowRicci solitonsKähler–Ricci solitonsPerelman entropyKähler geometryStability & uniquenessKähler–Einstein & K-stability