https://neckpinch.com/authors/wangjian-jian/New arXiv papers by Wangjian Jian, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2610.22941arxiv:2610.22941Fri, 16 Oct 2026 00:00:00 GMTWangjian Jian, Jian Song — In this paper, we establish geometric consequences of our Type I scalar curvature estimate for finite-time solutions of the Kähler-Ricci flow on compact Kähler manifolds. We prove a uniform global diameter bound and a Type I diameter bound for every fibre of finite time holomorphic contraction. These estimates yield Gromov-Hausdorff convergence to a unique compact metric space homeomorphic to the normal Kähler space determined by the limiting cohomology class. We also establish a local partial C⁰ estimate for the normalized flow and show that the tangent shrinkers are normal complex analytic varieties with log terminal singularities.Ricci flowKähler–Ricci flowRicci solitonsSingularitiesKähler geometryScalar curvatureLimit spaces & RCDAnalytic minimal model program https://arxiv.org/abs/2609.39620arxiv:2609.39620Wed, 30 Sep 2026 00:00:00 GMTWangjian Jian, Jian Song — We establish the Type I estimate for the scalar curvature of the Kahler-Ricci flow on compact Kahler manifolds, for arbitrary smooth initial metrics, whenever the flow develops finite time singularities. This estimate is a building block for the analytic minimal model program with Ricci flow.Ricci flowKähler–Ricci flowSingularitiesKähler geometryScalar curvatureAnalytic minimal model program https://arxiv.org/abs/2609.20513arxiv:2609.20513Thu, 17 Sep 2026 00:00:00 GMTWangjian Jian, Jian Song — The analytic minimal model program proposed in seeks to describe the birational transitions and fibre collapsing of the Kähler–Ricci flow through the geometry of its singularities. A fundamental conjecture of predicts Type I bounds for the scalar curvature and diameter of every fibre contratced by the limiting cohomological class at the finite singular time. This paper is a continuation of that establishes the Type I conjecture for collapsing solutions on Fano bundles with arbitrary smooth Fano fibres. If the fibre admits a smooth Kähler-Einstein metric, the Type I bound holds for the full curvature tensor.Ricci flowKähler–Ricci flowEinstein metricsSingularities(Non)collapsingKähler geometryScalar curvatureKähler–Einstein & K-stabilityAnalytic minimal model program https://arxiv.org/abs/2609.02878arxiv:2609.02878Wed, 02 Sep 2026 00:00:00 GMTWangjian Jian, Jian Song — We study collapsing finite-time singularities of the unnormalized Kähler–Ricci flow on Fano bundles arising in the analytic minimal model program. For a Fano bundle Xⁿ→ Y^m, we prove a maximal splitting theorem by the Kähler-Ricci flow that every tangent space at a fixed limiting point splits globally as (C^m,g_(rm E),J₀)×(Z',d',J'). If the fibre has complex dimension one, we prove a global Type-I bound for the full curvature tensor and show that the ambient and intrinsic diameters of every fibre are uniformly comparable to √(T-t). Furthermore, every tangent flow at any fixed limiting point is the round shrinking cylinder C^m×PP¹.Ricci flowKähler–Ricci flowSingularities(Non)collapsingKähler geometryAnalytic minimal model program 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/2305.09421arxiv:2305.09421Tue, 16 May 2023 00:00:00 GMTWangjian Jian, Yalong Shi — Assuming Perelman's estimates, we give a new proof of uniform L^∞ estimate along normalized Kähler-Ricci flow on Fano manifolds with Kähler-Einstein metrics, using Chen-Cheng's auxiliary Monge-Ampère equation and the Alexandrov-Bakelman-Pucci maximum principle. This proof does not use pluripotential theory.Ricci flowKähler–Ricci flowEinstein metricsKähler geometryKähler–Einstein & K-stability https://arxiv.org/abs/2202.06185arxiv:2202.06185Sun, 13 Feb 2022 00:00:00 GMTMax Hallgren, Wangjian Jian — We improve the description of F-limits of noncollapsed Ricci flows in the Kähler setting. In particular, the singular strata 𝓢^k of such metric flows satisfy 𝓢^(2j)=𝓢^(2j+1). We also prove an analogous result for quantitative strata, and show that any tangent flow admits a nontrivial one-parameter action by isometries, which is locally free on the cone link in the static case. The main results are established using parabolic regularizations of conjugate heat kernel potential functions based at almost-selfsimilar points, which may be of independent interest.Ricci flowSingularities(Non)collapsingHeat kernelKähler geometry https://arxiv.org/abs/2107.12515arxiv:2107.12515Mon, 26 Jul 2021 00:00:00 GMTWangjian Jian — In this note we derive an improved no-local-collapsing theorem of Ricci flow under the scalar curvature bound condition along the worldline of the basepoint. It is a refinement of Perelman's no-local-collapsing theorem.Ricci flow(Non)collapsingScalar curvature