https://neckpinch.com/authors/chung-jun-tsai/New arXiv papers by Chung-Jun Tsai, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2606.28767arxiv:2606.28767Sat, 27 Jun 2026 00:00:00 GMTPing-Hung Lee, Chung-Jun Tsai — We construct infinite-time singularities with vanishing mean curvature for Lagrangian mean curvature flow in Gibbons–Hawking spaces. We consider circle-invariant Lagrangian 2-spheres whose quotient curves are concave and are C²-close to a collection of consecutive collinear segments. We prove that the corresponding flow exists smoothly for all time and converges to the associated Aₙ₋₁-chain of special Lagrangian spheres. Although the mean curvature converges uniformly to zero, the second fundamental form becomes unbounded. More precisely, logmax |A( · ,t)| is comparable to √t as t→∞. The proof is based on a one-parameter family of barrier curves and a detailed analysis of their asymptotics. In this way, we refine the infinite-time convergence picture arising in the work of Lotay and Oliveira by proving curvature blow-up and estimating its rate in this semi-stable case.Mean curvature flowLagrangian mean curvature flowSingularities https://arxiv.org/abs/2406.04602arxiv:2406.04602Fri, 07 Jun 2024 00:00:00 GMTPing-Hung Lee, Chung-Jun Tsai — It is known that minimal Lagrangians in Kähler–Einstein manifolds of non-positive scalar curvature are linearly stable under Hamiltonian deformations. We prove that they are also stable under the Lagrangian mean curvature flow, and therefore establish the equivalence between linear stability and dynamical stability. Specifically, if one starts the mean curvature flow with a Lagrangian which is C¹-close and Hamiltonian isotopic to a minimal Lagrangian, the flow exists smoothly for all time, and converges to that minimal Lagrangian. Due to the work of Neves [Ann. of Math. 2013], this cannot be true for C⁰-closeness.Mean curvature flowLagrangian mean curvature flowEinstein metricsKähler geometryScalar curvatureStability & uniquenessKähler–Einstein & K-stability https://arxiv.org/abs/2401.02228arxiv:2401.02228Thu, 04 Jan 2024 00:00:00 GMTWei-Bo Su, Chung-Jun Tsai, Albert Wood — In this paper, we construct solutions of Lagrangian mean curvature flow which exist and are embedded for all time, but form an infinite-time singularity and converge to an immersed special Lagrangian as t→∞. In particular, the flow decomposes the initial data into a union of special Lagrangians intersecting at one point. This result shows that infinite-time singularities can form in the Thomas–Yau `semi-stable' situation. A precise polynomial blow-up rate of the second fundamental form is also shown. The infinite-time singularity formation is obtained by a perturbation of an approximate family N^(ε(t)) constructed by gluing in special Lagrangian `Lawlor necks' of size ε(t), where the dynamics of the neck size ε(t) are driven by the obstruction for the existence of nearby special Lagrangians to N^(ε(t)). This is inspired by the work of Brendle and Kapouleas regarding ancient solutions of the Ricci flow.Ricci flowMean curvature flowLagrangian mean curvature flowAncient solutionsSingularities https://arxiv.org/abs/2309.09432arxiv:2309.09432Mon, 18 Sep 2023 00:00:00 GMTChung-Jun Tsai, Mao-Pei Tsui, Mu-Tao Wang — Given an entire C² function u on ℝⁿ, we consider the graph of D u as a Lagrangian submanifold of ℝ²ⁿ, and deform it by the mean curvature flow in ℝ²ⁿ. This leads to the special Lagrangian evolution equation, a fully nonlinear Hessian type PDE. We prove long-time existence and convergence results under a 2-positivity assumption of (I+(D² u)²)⁻¹D² u. Such results were previously known only under the stronger assumption of positivity of D² u.Mean curvature flowLagrangian mean curvature flowLong-time existence & convergence https://arxiv.org/abs/2302.02512arxiv:2302.02512Mon, 06 Feb 2023 00:00:00 GMTChung-Jun Tsai, Mao-Pei Tsui, Mu-Tao Wang — We prove regularity, global existence, and convergence of Lagrangian mean curvature flows in the two-convex case. Such results were previously only known in the convex case, of which the current work represents a significant improvement. The proof relies on a newly discovered monotone quantity that controls two-convexity. Through a unitary transformation, same result for the mean curvature flow of area-decreasing Lagrangian submanifolds were established.Mean curvature flowLagrangian mean curvature flowLong-time existence & convergence https://arxiv.org/abs/2301.09222arxiv:2301.09222Sun, 22 Jan 2023 00:00:00 GMTChung-Jun Tsai, Mao-Pei Tsui, Mu-Tao Wang — A new monotone quantity in graphical mean curvature flows of higher codimensions is identified in this work. The submanifold deformed by the mean curvature flow is the graph of a map between Riemannian manifolds, and the quantity is monotone increasing under the area-decreasing condition of the map. The flow provides a natural homotopy of the corresponding map and leads to sharp criteria regarding the homotopic class of maps between complex projective spaces, and maps from spheres to complex projective spaces, among others.Mean curvature flow