https://neckpinch.com/authors/gang-tian/New arXiv papers by Gang Tian, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT 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/2310.08610arxiv:2310.08610Wed, 11 Oct 2023 00:00:00 GMTMax Hallgren, Wangjian Jian, Jian Song, Gang Tian — We establish geometric regularity for Type I blow-up limits of the Kähler-Ricci flow based at any sequence of Ricci vertices. As a consequence, the limiting flow is continuous in time in both Gromov-Hausdorff and Gromov-W₁ distance. In particular, the singular sets of each time slice and its tangent cones are close and of codimension no less than 4.Ricci flowKähler–Ricci flowSingularitiesKähler geometryLimit spaces & RCD https://arxiv.org/abs/2310.07945arxiv:2310.07945Wed, 11 Oct 2023 00:00:00 GMTWangjian Jian, Jian Song, Gang Tian — We establish the scalar curvature and distance bounds, extending Perelman's work on the Fano Kähler-Ricci flow to general finite time solutions of the Kähler-Ricci flow. These bounds are achieved by our Li-Yau type and Harnack estimates for weighted Ricci potential functions of the Kähler-Ricci flow. We further prove that the Type I blow-ups of the finite time solution always sub-converge in Gromov-Hausdorff sense to an ancient solution on a family of analytic normal varieties with suitable choices of base points. As a consequence, the Type I diameter bound is proved for almost every fibre of collapsing solutions of the Kähler-Ricci flow on a Fano fibre bundle. We also apply our estimates to show that every solution of the Kähler-Ricci flow with Calabi symmetry must develop Type I singularities, including both cases of high codimensional contractions and fibre collapsing.Ricci flowKähler–Ricci flowAncient solutionsSingularities(Non)collapsingHarnack inequalitiesKähler geometryScalar curvatureLimit spaces & RCD https://arxiv.org/abs/2310.07943arxiv:2310.07943Wed, 11 Oct 2023 00:00:00 GMTWangjian Jian, Jian Song, Gang Tian — In this note, we give a new proof for Perelman's scalar curvature and diameter estimates for the Kähler-Ricci flow on Fano manifolds. The proof relies on a new Harnack estimate for a special family of functions in space-time. Our new approach initiates the work in for general finite time solutions of the Kähler-Ricci flow.Ricci flowKähler–Ricci flowSingularitiesHarnack inequalitiesKähler geometryScalar curvature https://arxiv.org/abs/2210.15263arxiv:2210.15263Thu, 27 Oct 2022 00:00:00 GMTMinghao Miao, Gang Tian — In this note, we show that the solution of Kähler-Ricci flow on every Fano threefold from the family No.2.23 in the Mori-Mukai's list develops type II singularity. In fact, we show that no Fano threefold from the family No.2.23 admits Kähler-Ricci soliton and the Gromov-Hausdorff limit of the Kähler-Ricci flow must be a singular ℚ-Fano variety. This gives new examples of Fano manifolds of the lowest dimension on which Kähler-Ricci flow develops type II singularity.Ricci flowKähler–Ricci flowRicci solitonsKähler–Ricci solitonsSingularities3-manifoldsKähler geometryLimit spaces & RCD 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/2202.04564arxiv:2202.04564Wed, 09 Feb 2022 00:00:00 GMTJeffrey Streets, Gang Tian — Using dynamical stability of symplectic curvature flow, we show that on a compact Calabi-Yau manifold, any small symplectic deformation of a Kähler form remains Kähler.Other curvature flowsRicci-flat & Calabi–YauKähler geometryStability & uniqueness 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 https://arxiv.org/abs/0809.4040arxiv:0809.4040Tue, 23 Sep 2008 00:00:00 GMTJohn Morgan, Gang Tian — This article is a sequel to the book `Ricci Flow and the Poincare Conjecture' by the same authors. Using the main results of that book we establish the Geometrization Conjecture for all compact, orientable three-manifolds following the approach indicated by Perelman in his preprints on the subject. This approach is to study the collapsed part of the manifold as time goes to infinity in a Ricci flow with surgery. The main technique for this study is the theory of Alexandrov spaces. This theory gives local models for the collapsed part of the manifold. These local models can be glued together to prove that the collapsed part of the manifold is a graph manifold with incompressible boundary. From this and previous results, geometrization follows easily.Ricci flowSurgeryPoincaré conjectureGeometrization3-manifoldsDiscrete & graph Ricci https://arxiv.org/abs/math/0607607arxiv:math/0607607Tue, 25 Jul 2006 00:00:00 GMTJohn W. Morgan, Gang Tian — This manuscript contains a detailed proof of the Poincare Conjecture. The arguments we present here are expanded versions of the ones given by Perelman in his three preprints posted in 2002 and 2003. This is a revised version taking in account the comments of the referees and others. It has been reformatted in the AMS book style.Ricci flowPoincaré conjecture