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Tang-Kai Lee

Publications and source records attributed to Tang-Kai Lee.

13 recordsLinked to original sources
arXiv2026

Almost isoclinic Lagrangian submanifolds

We introduce the almost isoclinic region in the oriented Lagrangian Grassmannian ${\rm Lag}^+(n)$ of $\mathbb{C}^n$, an intrinsic higher-dimensional analog of a natural convex region in ${\rm Lag}^+(2) \simeq \mathbb S^1\times \mathbb S^2$. 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 $\Lambda$ on this region and prove that $\log \Lambda $ 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 $\Lambda$ must be a Lagrangian $n$-plane.

math.DG↗
arXiv2026

Non-uniqueness of geodesic limits and a question of Grayson and Gage

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 $\mathbb{S}^2$ 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.

math.DG↗
arXiv2026

Closed mean curvature flows with prescribed tangent flows

Given an embedded shrinker $\Sigma$ in $\mathbb{R}^{n+1}$ that is either closed, asymptotically conical, or a Cartesian product of such a shrinker with $\mathbb{R}^k$, we construct a closed embedded mean curvature flow whose tangent flow at the first singularity is modeled on $\Sigma$. 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.

math.DG↗
arXiv2025

Local mollification of metrics with small curvature concentration

In this work, we establish a local smoothing result on metrics with small curvature concentration with respect to Sobolev constants and volume growth. In contrast with all previous works, we remove the Ricci curvature condition and completely localize the smoothing. As an application, we prove the compactness of the space of compact manifolds with bounded curvature concentration under Ahlfors $n$-regularity and bounded Sobolev constant. In the complete non-compact case, we show that manifolds with Euclidean type Sobolev inequality, Euclidean volume growth, and small curvature concentration are necessarily diffeomorphic to Euclidean spaces.

math.DG↗
arXiv2025

Arnold-Thom conjecture for the arrival time of surfaces

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 $\mathbb R^{n+1}$ 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 $C^2.$ The results also apply to mean curvature flows starting from two-spheres or generic closed surfaces.

math.DG↗
arXiv2025

An Intersection Principle for Mean Curvature Flow

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.

math.DG↗
arXiv2025

Planarity and convexity for pinched ancient solutions of mean curvature flow

We prove a parabolically scale-invariant variation of the planarity estimate in \cite{Na22} for higher codimension mean curvature flow, borrowing ideas from work of Brendle--Huisken--Sinestrari \cite{BHS}. 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.

math.DG↗
arXiv2024

Ancient caloric functions and parabolic frequency on graphs

We study ancient solutions to discrete heat equations on some weighted graphs. On a graph of the form of a product with $\bb Z,$ we show that there are no non-trivial ancient solutions with polynomial growth. This result is parallel to the case of finite graphs, which is also discussed. Along the way, we prove a backward uniqueness result for solutions with appropriate decaying rate based on a monotonicity formula of parabolic frequency.

math.AP↗
arXiv2024

Closed mean curvature flows with asymptotically conical singularities

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.

math.DG↗
arXiv2023

Compactness and rigidity of self-shrinking surfaces

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.

math.DG↗
arXiv2022

Parabolic frequency for the mean curvature flow

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.

math.DG↗
arXiv2021

Convexity of $λ$-hypersurfaces

We prove that any $n$-dimensional closed mean convex $λ$-hypersurface is convex if $λ\le 0.$ This generalizes Guang's work on $2$-dimensional strictly mean convex $λ$-hypersurfaces. As a corollary, we obtain a gap theorem for closed $λ$-hypersurfaces with $λ\le 0.$

math.DG↗