https://neckpinch.com/authors/max-hallgren/New arXiv papers by Max Hallgren, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2609.16733arxiv:2609.16733Tue, 15 Sep 2026 00:00:00 GMTCharles Cifarelli, Ronan Conlon, Max Hallgren, Junsheng Zhang — For any volume-collapsing finite-time singularity of a Kähler-Ricci flow on a compact Kähler surface, we show the flow satisfies a Type I curvature bound and classify the corresponding tangent flows. Combined with previous results, this shows that any finite-time singularity of a Kähler-Ricci flow on a compact Kähler surface is of Type I.Ricci flowKähler–Ricci flowSingularities(Non)collapsingKähler geometry https://arxiv.org/abs/2608.19152arxiv:2608.19152Wed, 19 Aug 2026 00:00:00 GMTLongteng Chen, Max Hallgren, Lucas Lavoyer — We describe certain finite-time singularities of the Kähler-Ricci flow arising in the analytic minimal model program. Assuming that convergence to an asymptotically conical Kähler-Ricci shrinker is realized by holomorphic maps, we prove that, in a fixed holomorphic gauge, the nearby flow is modeled on the shrinker at the level of Kähler potentials. Consequently, every noncollapsed Kähler-Ricci flow through singularities in complex dimension two is modeled on a shrinker-cone-expander transition, confirming a strong form of Song's conjectural picture. We also show analogous results in higher dimensions under the Calabi ansatz, and improve known results in the compact shrinker case. These give the first compact Ricci flows through conical singularities whose small-scale behavior is fully described.Ricci flowKähler–Ricci flowRicci solitonsSingularities(Non)collapsingKähler geometryWeak & singular flowsBoundaries & non-compactAnalytic minimal model program https://arxiv.org/abs/2605.25213arxiv:2605.25213Sun, 24 May 2026 00:00:00 GMTMax Hallgren, Junsheng Zhang — We prove that any singular Kähler–Ricci shrinker X arising as a noncollapsed limit of Kähler–Ricci flows admits a natural structure of a polarized Fano fibration. We also show that it is simply connected, has unique tangent cones at every point, and is a smooth orbifold outside a subset of complex codimension at least three. As an application, we prove a new long-time pseudolocality theorem for almost-selfsimilar Kähler–Ricci flows.Ricci flowKähler–Ricci flowRicci solitons(Non)collapsingKähler geometryAnalytic minimal model program https://arxiv.org/abs/2604.04223arxiv:2604.04223Sun, 05 Apr 2026 00:00:00 GMTLongteng Chen, Max Hallgren, Lucas Lavoyer — Let (Y,g₀) be a compact analytic space with a finite number of singular points, where the metric at each singular point is modelled on a Kähler cone with smooth canonical model. We show that the Kähler-Ricci flow with such initial data satisfies a C/t curvature bound, and that the flow near each singular point is modelled on the unique Kähler-Ricci expander asymptotic to the corresponding cone. Our motivation is to give a geometric description of the Kähler–Ricci flow emerging from singularities arising in the analytic minimal model program.Ricci flowKähler–Ricci flowRicci solitonsSingularitiesKähler geometryAnalytic minimal model program https://arxiv.org/abs/2509.14154arxiv:2509.14154Wed, 17 Sep 2025 00:00:00 GMTHarry Fluck, Max Hallgren — In this article we prove an ε-regularity theorem for non-collapsed Ricci flows, and use this to prove new estimates for singularity models of Fano Kähler-Ricci flows. In the course of our proof, we find a criterion for uniform convergence of solutions to the heat equation along a sequence of F-converging Ricci flows, and apply this to new parabolic regularizations of some natural geometric quantities.Ricci flowKähler–Ricci flowSingularities(Non)collapsingKähler geometry https://arxiv.org/abs/2505.01943arxiv:2505.01943Sat, 03 May 2025 00:00:00 GMTMax Hallgren, Gábor Székelyhidi — We study two different natural notions of singular Kähler-Einstein metrics on normal complex varieties. In the setting of singular Ricci flat Kähler cone metrics that arise as non-collapsed limits of sequences of Kähler-Einstein metrics or Kähler-Ricci flows, we show that an a priori weaker notion is equivalent to the stronger one introduced by Eyssidieux-Guedj-Zeriahi, and in particular the underlying variety has log terminal singularities in this case. Our method applies to more general singular Kähler-Einstein spaces as well, assuming that they define RCD spaces.Ricci flowKähler–Ricci flowEinstein metricsRicci-flat & Calabi–YauSingularities(Non)collapsingKähler geometryWeak & singular flowsLimit spaces & RCDKähler–Einstein & K-stability https://arxiv.org/abs/2502.19804arxiv:2502.19804Thu, 27 Feb 2025 00:00:00 GMTRonan J. Conlon, Max Hallgren, Zilu Ma — We show that any non-collapsed finite time singularity of the Ricci flow on a compact Kähler surface is of Type I. Combined with a previous result of the first author, Cifarelli, and Deruelle, it follows that any such singularity is modeled on the shrinking Ricci soliton of Feldman-Ilmanen-Knopf on the total space of the line bundle 𝓞_(ℙ¹)(−1)→ℙ¹.Ricci flowRicci solitonsSingularities(Non)collapsingKähler geometry https://arxiv.org/abs/2502.09825arxiv:2502.09825Thu, 13 Feb 2025 00:00:00 GMTYifan Chen, Shih-Kai Chiu, Max Hallgren, Gábor Székelyhidi, Tat Dat Tô, Freid Tong — We show that a general class of singular Kähler metrics with Ricci curvature bounded below define Kähler currents. In particular the result applies to singular Kähler-Einstein metrics on klt pairs, and an analogous result holds for Kähler-Ricci solitons. In addition we show that if a singular Kähler-Einstein metric can be approximated by smooth metrics on a resolution whose Ricci curvature has negative part that is bounded uniformly in L^p for p > (2n-1)/n, then the metric defines an RCD space.Ricci solitonsKähler–Ricci solitonsEinstein metricsKähler geometryLimit spaces & RCDKähler–Einstein & K-stabilityAnalytic minimal model program https://arxiv.org/abs/2312.06577arxiv:2312.06577Mon, 11 Dec 2023 00:00:00 GMTMax Hallgren — We show that any tangent cone of a singular shrinking Kähler-Ricci soliton is a normal affine algebraic variety. Moreover, the regular set of such a tangent cone in the metric sense coincides with the regular set in the algebraic sense. Along the way, we give a parabolic proof of Hörmander's L² estimate, which can be used to solve the ∂-equation on any singular shrinking Kähler-Ricci soliton.Ricci solitonsKähler–Ricci solitonsSingularitiesKähler 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/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/2201.09387arxiv:2201.09387Sun, 23 Jan 2022 00:00:00 GMTTimothy Buttsworth, Maximilien Hallgren, Yongjia Zhang — We construct a rotationally invariant Ricci flow through surgery starting at any closed rotationally invariant Riemannian manifold. We demonstrate that a sequence of such Ricci flows with surgery converges to a Ricci flow spacetime in the sense of [32]. Results of Bamler-Kleiner [8] and Haslhofer [29] then guarantee the uniqueness and stability of these spacetimes given initial data. We simplify aspects of this proof in our setting, and show that for rotationally invariant Ricci flows, the closeness of spacetimes can be measured by equivariant comparison maps. Finally we show that the blowup rate of the curvature near a singular time for these Ricci flows is bounded by the inverse of remaining time squared.Ricci flowSingularitiesSurgeryStability & uniquenessRotational symmetry & warped products