https://neckpinch.com/authors/man-chun-lee/New arXiv papers by Man-Chun Lee, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2606.23272arxiv:2606.23272Mon, 22 Jun 2026 00:00:00 GMTMan-Chun Lee, Florian Litzinger, Miles Simon — In this work, we study the positive mass theorem under critical low regularity assumptions using Ricci flow smoothing. We show that asymptotically flat manifolds (Mⁿ,g) of regularity L^∞∩ W^(1,n) with non-negative distributional scalar curvature have non-negative ADM mass. Furthermore, when the ADM mass vanishes, the manifold is globally isometric to Euclidean space with respect to an integral distance introduced by De Cecco-Palmieri. This extends the recent work of Hafemann to the critical regularity case. Our approach is based on showing that Riemannian metrics of regularity L^∞∩ W^(1,n), whose scalar curvature is bounded from below in the distributional sense, admit a Ricci flow smoothing whose scalar curvature is bounded from below by the same initial lower bound in the classical sense. In contrast, Cecchini-Frenck-Zeidler constructed examples of metrics which are in L^∞∩ W^(1,p) for all 2<p<n, and whose distributional scalar curvature is bounded from below, that cannot be approximated by smooth metrics with the same scalar curvature lower bound. In this sense, our result is optimal.Ricci flowScalar curvatureBoundaries & non-compactPhysics https://arxiv.org/abs/2603.22086arxiv:2603.22086Mon, 23 Mar 2026 00:00:00 GMTAlix Deruelle, Man-Chun Lee, Felix Schulze, Miles Simon, Peter M. Topping — Hamilton's pinching conjecture, that three-dimensional complete non-compact manifolds with pinched Ricci curvature are flat, has recently been resolved using Ricci flow. In this paper we prove a direct analogue of that result in all dimensions. In order to do so we develop a lifting technique that allows us to handle manifolds that are collapsed at infinity. This new method also gives an alternative way of handling collapsed manifolds in the known three-dimensional case. As part of this approach, we prove a Ricci flow curvature estimate of a type that would normally be derived from the Harnack inequality, but without requiring the strong curvature positivity hypothesis demanded by Harnack. We give an improved gap theorem as a further application.Ricci flowHarnack inequalitiesCurvature pinching3-manifoldsSphere theoremsBoundaries & non-compact https://arxiv.org/abs/2602.19913arxiv:2602.19913Mon, 23 Feb 2026 00:00:00 GMTMan-Chun Lee, Valentino Tosatti, Junsheng Zhang — We show that the normalized Kähler-Ricci flow on a compact Kähler manifold with semiample canonical bundle converges in the Gromov-Hausdorff topology to the metric completion of the twisted Kähler-Einstein metric on the canonical model, as conjectured by Song-Tian's analytic mimimal model program.Ricci flowKähler–Ricci flowEinstein metricsKähler geometryLimit spaces & RCDLong-time existence & convergenceKähler–Einstein & K-stabilityAnalytic minimal model program https://arxiv.org/abs/2510.23005arxiv:2510.23005Mon, 27 Oct 2025 00:00:00 GMTPak-Yeung Chan, Yi Lai, Man-Chun Lee — For any n≥ 4, we construct an (n-2)-parameter family of steady gradient Ricci solitons with non-negative curvature operator and prescribed by the eigenvalues of Ricci tensor at a critical point of the soliton potential. Among them lies an (n-3)-parameter subfamily of non-collapsed solitons. These solitons generalized the flying wings constructed by the second named author and produced new examples of steady gradient Ricci solitons with non-negative curvature operator for n≥ 4. Our approach is based on constructing continuous families of Ricci flows smoothing emanating from continuous families of spherical polyhedra which still preserves symmetry. This is built upon a new stability result of Ricci flows with scaling invariant estimates. As another application of the method, we prove the stability of asymptotically conical expanding solitons constructed by Deruelle under L^∞ perturbation of links. In particular, the C⁰-convergence of smooth links implies the smooth convergence of the expanding solitons.Ricci flowRicci solitons(Non)collapsingStability & uniquenessBoundaries & non-compactEigenvalues & spectrum https://arxiv.org/abs/2504.08189arxiv:2504.08189Fri, 11 Apr 2025 00:00:00 GMTMing Hsiao, Man-Chun Lee — In this work, we study a gap phenomenon in locally conformally flat Riemannian manifolds with non-negative Ricci curvature. We construct complete solutions to the Yamabe flow that exhibit instantaneous bounded curvature as they evolve. Using this, we demonstrate that if the curvature decays quickly enough in an integral sense, then the manifold must be flat. This partially generalizes the results of Chen-Zhu and Ma.Yamabe flow https://arxiv.org/abs/2503.20292arxiv:2503.20292Wed, 26 Mar 2025 00:00:00 GMTMan-Chun Lee — In this work, we prove uniqueness for complete non-compact Ricci flow with scaling invariant curvature bound. This generalizes the earlier work of Chen-Zhu, Kotschwar and covers most of the example of Ricci flows with unbounded curvature. In dimension three, we use it to show that complete Ricci flow starting from uniformly non-collapsed, non-negatively curved manifold is unique, extending the strong uniqueness Theorem of Chen. This is based on solving Ricci-harmonic map heat flow in unbounded curvature background.Ricci flow(Non)collapsing3-manifoldsStability & uniquenessBoundaries & non-compactHarmonic maps & DeTurck trick https://arxiv.org/abs/2503.12416arxiv:2503.12416Sun, 16 Mar 2025 00:00:00 GMTPak-Yeung Chan, Man-Chun Lee — In this work, we construct several sequences of metrics on sphere with different limiting behaviors. By combining with the work of Deruelle, we use it and the localized maximum principle to construct various examples of expanding gradient Ricci solitons with positive curvature and exotic curvature decay. This answers a question proposed by Chow-Lu-Ni and also a question by Cao-Liu, respectively.Ricci solitons https://arxiv.org/abs/2407.06575arxiv:2407.06575Tue, 09 Jul 2024 00:00:00 GMTMan-Chun Lee, Stephen Shang Yi Liu — In this work, we study the short-time existence theory of Ricci-DeTurck flow starting from rough metrics which satisfy a Morrey-type integrability condition. Using the rough existence theory, we show the preservation and improvement of distributional scalar curvature lower bounds provided the singular set for such metrics is not too large. As an application, we use the Ricci flow smoothing to study the removable singularity for scalar curvature rigidity in the compact case under Morrey regularity conditions. Our result supplements those of Jiang-Sheng-Zhang.Ricci flowSingularitiesScalar curvatureWeak & singular flowsStability & uniquenessHarmonic maps & DeTurck trick https://arxiv.org/abs/2405.04208arxiv:2405.04208Tue, 07 May 2024 00:00:00 GMTHans-Joachim Hein, Man-Chun Lee, Valentino Tosatti — We consider the Kähler-Ricci flow on compact Kähler manifolds with semiample canonical bundle and intermediate Kodaira dimension, and show that the flow collapses to a canonical metric on the base of the Iitaka fibration in the locally smooth topology and with bounded Ricci curvature away from the singular fibers. This follows from an asymptotic expansion for the evolving metrics, in the spirit of recent work of the first and third-named authors on collapsing Calabi-Yau metrics, and proves two conjectures of Song and Tian.Ricci flowKähler–Ricci flowRicci-flat & Calabi–Yau(Non)collapsingKähler geometryLong-time existence & convergenceAnalytic minimal model program https://arxiv.org/abs/2404.12755arxiv:2404.12755Fri, 19 Apr 2024 00:00:00 GMTPak-Yeung Chan, Man-Chun Lee, Luke T. Peachey — Motivated by recent work of Deruelle-Schulze-Simon, we study complete weakly PIC1 Ricci flows with Euclidean volume growth coming out of metric cones. We show that such a Ricci flow must be an expanding gradient Ricci soliton, and as a consequence, any metric cone at infinity of a complete weakly PIC1 Kähler manifold with Euclidean volume growth is biholomorphic to complex Euclidean space in a canonical way.Ricci flowRicci solitonsSphere theoremsKähler geometryWeak & singular flows https://arxiv.org/abs/2312.10940arxiv:2312.10940Mon, 18 Dec 2023 00:00:00 GMTMan-Chun Lee, Luen-Fai Tam, Jingbo Wan — In this work, we consider the area non-increasing map between manifolds with positive curvature. By exploring the strong maximum principle along the graphical mean curvature flow, we show that an area non-increasing map between certain positively curved manifolds is either homotopy trivial, Riemannian submersion, local isometry or isometric immersion. This implies that an area non-increasing self map of ℂP}ⁿ, n≥ 2 is either homotopically trivial or is an isometry. This confirms a speculation of Tsai-Tsui-Wang. We also use Brendle's sphere Theorem and mean curvature flow coupled with Ricci flow to establish related results on manifolds with positive 1-isotropic curvature.Ricci flowMean curvature flowSphere theoremsStability & uniqueness https://arxiv.org/abs/2305.01396arxiv:2305.01396Tue, 02 May 2023 00:00:00 GMTPak-Yeung Chan, Man-Chun Lee — In this work, we use the Ricci flow approach to study the gap phenomenon of Riemannian manifolds with non-negative curvature and sub-critical scaling invariant curvature decay. The first main result is a quantitative Ricci flow existence theory without non-collapsing assumption. We use it to show that complete non-compact manifolds with non-negative complex sectional curvature and sufficiently small average curvature decay are necessarily flat. The second main result concerns three-manifolds with non-negative Ricci curvature of quadratic decay. By combining our newly established curvature estimate and method in Kähler geometry, we show that if the curvature decays slightly faster even in average sense, the manifold must be flat. This strengthens a result of Reiris. In the compact case, we use the Ricci flow regularization to generalize the celebrated Gromov-Ruh Theorem in this direction.Ricci flow(Non)collapsing3-manifoldsKähler geometryBoundaries & non-compact https://arxiv.org/abs/2304.02914arxiv:2304.02914Thu, 06 Apr 2023 00:00:00 GMTMan-Chun Lee — In this work, we extend the existence theory of non-collapsed Ricci flows from point-wise curvature lower bound to Kato-type lower bound. As an application, we prove that compact three dimensional non-collapsed strong Kato limit space is homeomorphic to a smooth manifold. The result also holds in higher dimension under uniform strong Kato lower bound of 1-isotropic curvature. We also use the Ricci flow smoothing to study stability problem in scalar curvature geometry.Ricci flow(Non)collapsing3-manifoldsScalar curvatureStability & uniqueness https://arxiv.org/abs/2211.07623arxiv:2211.07623Mon, 14 Nov 2022 00:00:00 GMTMan-Chun Lee, Peter M. Topping — Recently it has been proved (Lee-Topping 2022, Deruelle-Schulze-Simon 2022, Lott 2019) that three-dimensional complete manifolds with non-negatively pinched Ricci curvature must be flat or compact, thus confirming a conjecture of Hamilton. In this paper we generalise our work on the existence of Ricci flows from non-compact pinched three-manifolds in order to prove a higher-dimensional analogue. We construct a solution to Ricci flow, for all time, starting with an arbitrary complete non-compact manifold that is PIC1 pinched. As an application we prove that any complete manifold of non-negative complex sectional curvature that is PIC1 pinched must be flat or compact.Ricci flow3-manifoldsSphere theoremsBoundaries & non-compact https://arxiv.org/abs/2207.11017arxiv:2207.11017Fri, 22 Jul 2022 00:00:00 GMTMan-Chun Lee, Luen-Fai Tam — Motivated by the Lipschitz rigidity problem in scalar curvature geometry, we prove that if a closed smooth spin manifold admits a distance decreasing continuous map of non-zero degree to a sphere, then either the scalar curvature is strictly less than the sphere somewhere or the map is a distance isometry. Moreover, the property also holds for continuous metrics with scalar curvature lower bound in some weak sense. This extends a result in the recent work of Cecchini-Hanke-Schick and answers a question of Gromov. The method is based on studying the harmonic map heat flow coupled with the Ricci flow from rough initial data to reduce the case to smooth metrics and smooth maps so that results by Llarull can be applied.Ricci flowScalar curvatureWeak & singular flowsStability & uniquenessHarmonic maps & DeTurck trick https://arxiv.org/abs/2207.07495arxiv:2207.07495Fri, 15 Jul 2022 00:00:00 GMTPak-Yeung Chan, Shaochuang Huang, Man-Chun Lee — In this work, we construct distance like functions with integral hessian bound on manifolds with small curvature concentration and use it to construct Ricci flows on manifolds with possibly unbounded curvature. As an application, we study the geometric structure of those manifolds without bounded curvature assumption. In particular, we show that manifolds with Ricci lower bound, non-negative scalar curvature, bounded entropy, Ahlfors n-regular and small curvature concentration are topologically Euclidean.Ricci flowScalar curvature https://arxiv.org/abs/2204.00504arxiv:2204.00504Fri, 01 Apr 2022 00:00:00 GMTMan-Chun Lee, Peter M. Topping — We show that every complete non-compact three-manifold with non-negatively pinched Ricci curvature admits a complete Ricci flow solution for all positive time, with scale-invariant curvature decay and preservation of pinching. Combining with recent work of Lott and Deruelle-Schulze-Simon gives a proof of Hamilton's pinching conjecture without additional hypotheses.Ricci flowCurvature pinching3-manifoldsBoundaries & non-compact https://arxiv.org/abs/2202.12564arxiv:2202.12564Fri, 25 Feb 2022 00:00:00 GMTMan-Chun Lee, Peter M. Topping — We consider the problem of when a smooth Ricci flow, for positive time, that attains smooth initial data in a weak sense must be smooth down to the initial time. We obtain curvature estimates for an example where this fails. We prove a positive result in the case that the flow satisfies a lower IC1 curvature bound, equivalent to a lower Ricci bound in three dimensions. As an application, we prove that Gromov-Hausdorff limits of WPIC1 manifolds are WPIC1.Ricci flowLimit spaces & RCD