https://neckpinch.com/authors/tang-kai-lee/New arXiv papers by Tang-Kai Lee, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2609.19648arxiv:2609.19648Thu, 17 Sep 2026 00:00:00 GMTKahnrad Braxton, Tang-Kai Lee, Jonathan J. Zhu — We prove the linear stability of an embedded mean curvature flow shrinker in ℝ⁴ with the topology of the Möbius bundle. This provides a non-flat, non-spherical stable shrinker in higher codimension, in sharp contrast with the classification of stable hypersurface shrinkers. Topologically, the Möbius shrinker models the reversal of a real blow-up, suggesting that stable higher-codimension singularities may encode non-trivial topological operations under mean curvature flow.Mean curvature flowRicci solitonsSelf-shrinkers & translatorsSingularitiesStability & uniqueness https://arxiv.org/abs/2608.14444arxiv:2608.14444Fri, 14 Aug 2026 00:00:00 GMTTang-Kai Lee, Mu-Tao Wang — We introduce the almost isoclinic region in the oriented Lagrangian Grassmannian rm Lag⁺(n) of ℂⁿ, an intrinsic higher-dimensional analog of a natural convex region in rm Lag⁺(2) ≃ S¹× S². A Lagrangian submanifold is called almost isoclinic if its Gauss map takes values in this region, extending the graphical condition that the characteristic angles of the tangent plane remain uniformly close. We construct a canonical positive function Λ on this region and prove that log Λ is concave with respect to the invariant Grassmannian metric. This property yields subharmonicity and monotonicity formulas for minimal Lagrangians and Lagrangian mean curvature flow. As applications, we prove rigidity and Bernstein-type results, including that a complete connected almost isoclinic minimal Lagrangian with a positive lower bound for Λ must be a Lagrangian n-plane.Mean curvature flowLagrangian mean curvature flowStability & uniqueness https://arxiv.org/abs/2608.04855arxiv:2608.04855Wed, 05 Aug 2026 00:00:00 GMTShrey Aryan, Tang-Kai Lee — Grayson and Gage proved that an immortal curve shortening flow of simple closed curves on a closed surface converges subsequentially to a closed geodesic, and they asked whether this limiting geodesic is unique. We answer this question negatively by constructing a smooth Riemannian metric on S² and an immortal simple closed curve shortening flow that converges along different sequences of times to every geodesic in a one-parameter family of distinct simple closed geodesics.Curve shorteningStability & uniquenessLong-time existence & convergence https://arxiv.org/abs/2607.23879arxiv:2607.23879Sun, 26 Jul 2026 00:00:00 GMTJingwen Chen, Tang-Kai Lee, Ao Sun, Xinrui Zhao — Given an embedded shrinker Σ in ℝⁿ⁺¹ that is either closed, asymptotically conical, or a Cartesian product of such a shrinker with ℝ^k, we construct a closed embedded mean curvature flow whose tangent flow at the first singularity is modeled on Σ. We also prescribe the first-order asymptotics of the tangent flow. This result is a consequence of a more general theorem that allows us to construct mean curvature flows with an additional force whose tangent flow and first-order asymptotics at the first singularity are prescribed.Mean curvature flowRicci solitonsSingularitiesBoundaries & non-compact https://arxiv.org/abs/2505.11600arxiv:2505.11600Fri, 16 May 2025 00:00:00 GMTTang-Kai Lee, Alec Payne — The avoidance principle says that mean curvature flows of hypersurfaces remain disjoint if they are disjoint at the initial time. We prove several generalizations of the avoidance principle that allow for intersections of hypersurfaces. First, we prove that the Hausdorff dimension of the intersection of two mean curvature flows is non-increasing over time, and we find precise information on how the dimension changes. We then show that the self-intersection of an immersed mean curvature flow has non-increasing dimension over time. Next, we extend the intersection dimension monotonicity to Brakke flows and level set flows which satisfy a localizability condition, and we provide examples showing that the monotonicity fails for general weak solutions. We find a localization result for level set flows with finitely many singularities, and as a consequence, we obtain a fattening criterion for these flows which depends on the behavior of intersections with smooth flows.Mean curvature flowSingularitiesWeak & singular flows https://arxiv.org/abs/2504.17922arxiv:2504.17922Thu, 24 Apr 2025 00:00:00 GMTTang-Kai Lee, Keaton Naff, Jingze Zhu — We prove a parabolically scale-invariant variation of the planarity estimate in for higher codimension mean curvature flow, borrowing ideas from work of Brendle–Huisken–Sinestrari. Additionally, we prove convexity for pinched complete ancient solutions of the mean curvature flow in codimension one. Then we put these estimates together to characterize certain pinched complete ancient solutions and shrinkers in higher codimension. We include some discussion of future research directions in this area of mean curvature flow.Mean curvature flowRicci solitonsAncient solutions https://arxiv.org/abs/2405.19064arxiv:2405.19064Wed, 29 May 2024 00:00:00 GMTTang-Kai Lee, Jingze Zhu — Following Łojasiewicz's uniqueness theorem and Thom's gradient conjecture, Arnold proposed a stronger version about the existence of limit tangents of gradient flow lines for analytic functions. We prove Łojasiewicz's theorem and Arnold's conjecture in the context of arrival time functions for mean curvature flows in ℝⁿ⁺¹ with neck or non-degenerate cylindrical singularities. In particular, we prove the conjectures for all mean convex mean curvature flows of surfaces, including the cases when the arrival time functions are not C². The results also apply to mean curvature flows starting from two-spheres or generic closed surfaces.Mean curvature flowSingularitiesStability & uniqueness https://arxiv.org/abs/2405.15577arxiv:2405.15577Fri, 24 May 2024 00:00:00 GMTTang-Kai Lee, Xinrui Zhao — In this paper, we prove that for any asymptotically conical self-shrinker, there exists an embedded closed hypersurface such that the mean curvature flow starting from it develops a singularity modeled on the given shrinker. The main technique is the Ważewski box argument, used by Stolarski in the proof of the corresponding theorem in the Ricci flow case. As a corollary, our construction, combined with the works of Angenent–Ilmanen–Velázquez and Chodosh–Daniels-Holgate–Schulze, implies the existence of fattening level set flows starting from smooth embedded closed hypersurfaces. These provide examples related to a question asked by Evans–Spruck.Ricci flowMean curvature flowRicci solitonsSelf-shrinkers & translatorsSingularitiesWeak & singular flowsBoundaries & non-compact https://arxiv.org/abs/2301.10883arxiv:2301.10883Thu, 26 Jan 2023 00:00:00 GMTTang-Kai Lee, Xinrui Zhao — In this paper, we prove the uniqueness of asymptotically conical tangent flows in all codimensions. This is based on an early work of Chodosh-Schulze, who proved the uniqueness in the hypersurface case.Mean curvature flowSingularitiesStability & uniquenessBoundaries & non-compact https://arxiv.org/abs/2210.14286arxiv:2210.14286Tue, 25 Oct 2022 00:00:00 GMTJulius Baldauf, Tang-Kai Lee — This paper defines a parabolic frequency for solutions of the heat equation along homothetically shrinking mean curvature flows and proves its monotonicity along such flows. As a corollary, frequency monotonicity provides a proof of backwards uniqueness. Additionally, for solutions of more general parabolic equations on mean curvature flow shrinkers, this paper provides bounds on the derivative of the frequency, which similarly imply backwards uniqueness.Mean curvature flowStability & uniqueness https://arxiv.org/abs/2108.03919arxiv:2108.03919Mon, 09 Aug 2021 00:00:00 GMTTang-Kai Lee — The entropy functional introduced by Colding and Minicozzi plays a fundamental role in the analysis of mean curvature flow. However, unlike the hypersurface case, relatively little about the entropy is known in the higher-codimension case. In this note, we use measure-theoretical techniques and rigidity results for self-shrinkers to prove a compactness theorem for a family of self-shrinking surfaces with low entropy. Based on this, we prove the existence of entropy minimizers among self-shrinking surfaces and improve some rigidity results.Mean curvature flowSelf-shrinkers & translatorsLimit spaces & RCDStability & uniqueness