https://neckpinch.com/authors/jian-song/New arXiv papers by Jian Song, 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/2209.09428arxiv:2209.09428Tue, 20 Sep 2022 00:00:00 GMTBin Guo, Duong H. Phong, Jian Song, Jacob Sturm — Diameter estimates for Kähler metrics are established which require only an entropy bound and no lower bound on the Ricci curvature. The proof builds on recent PDE techniques for L^∞ estimates for the Monge-Ampère equation, with a key improvement allowing degeneracies of the volume form of codimension strictly greater than one. As a consequence, diameter bounds are obtained for long-time solutions of the Kähler-Ricci flow and finite-time solutions when the limiting class is big, as well as for special fibrations of Calabi-Yau manifolds.Ricci flowKähler–Ricci flowRicci-flat & Calabi–YauSingularitiesKähler geometryLong-time existence & convergence