https://neckpinch.com/authors/kyeongsu-choi/New arXiv papers by Kyeongsu Choi, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2609.08112arxiv:2609.08112Tue, 08 Sep 2026 00:00:00 GMTKyeongsu Choi, Panagiota Daskalopoulos, Natasa Sesum — In this paper, we study the formation, precise asymptotics, and structural stability of neckpinch singularities in mean curvature flow. Motivated by the static rigidity of cylindrical self-shrinkers established by Colding, Ilmanen and Minicozzi, we first prove a dynamical rigidity result: mean curvature flow of hypersurfaces that are initially graphically close to a generalized cylinder on a sufficiently large scale, subject to a localized quadratic upward bending, inevitably develop a neckpinch singularity in finite time. We also establish sharp asymptotic expansions for the profile functions of these locally evolving graphs. We show that the rescaled graphical radius converges to a specific polynomial profile with quadratic bending, proving that the resulting singularities are nondegenerate. Finally, we establish an openness theorem showing that nondegenerate neckpinches are structurally stable under C^2 perturbations of the initial data. Combined with recent density theorems for the 3-dimensional mean curvature flow by Szekelyhidi, our results confirm that nondegenerate neckpinches constitute a generic and stable phenomenon in 3-dimensional mean curvature flow.Mean curvature flowSelf-shrinkers & translatorsSingularities3-manifoldsStability & uniqueness https://arxiv.org/abs/2608.30787arxiv:2608.30787Mon, 31 Aug 2026 00:00:00 GMTKyeongsu Choi, Robert Haslhofer — In a recent breakthrough, Bamler-Lai proved the mean-convex neighborhood conjecture in all dimensions by showing that any nontrivial ancient asymptotically cylindrical mean curvature flow is – up to splitting Euclidean factors – either a translating bowl, or an ancient oval, or a translating oval-bowl. In this paper, we provide a short alternative argument for two of the three scenarios. Specifically, using more elementary/traditional methods, we show that if the convergence to the round cylinder is fast then the solution is a bowl times a Euclidean factor, and if the convergence is slow then the solution is an ancient oval. Moreover, the present paper also yields a new proof of the mean-convex neighborhood conjecture for neck-singularities that substantially simplifies and streamlines the approach from our prior work joint with Hershkovits (Acta '22) and Hershkovits-White (Inventiones '22).Mean curvature flowAncient solutionsSingularities https://arxiv.org/abs/2603.09112arxiv:2603.09112Tue, 10 Mar 2026 00:00:00 GMTKyeongsu Choi, Dong-Hwi Seo, Wei-Bo Su, Kai-Wei Zhao — We prove that any ancient smooth embedded finite-entropy curve shortening flow is one of the following: a static line, a shrinking circle, a paper clip, a translating grim reaper, or a graphical ancient trombone. An ancient trombone is an immersed ancient flow, either compact or non-compact, obtained by gluing together m translating grim reaper curves. For each m, there exists a (2m-1)-parameter family of graphical ancient trombones, up to rigid motions and time shifts as constructed by Angenent-You. In particular, our result implies that any compact ancient smooth embedded finite-entropy flow is convex. Moreover, any non-compact ancient smooth embedded finite-entropy flow is either a static line or a complete graph over a fixed open interval.Curve shorteningAncient solutionsSelf-shrinkers & translatorsBoundaries & non-compact https://arxiv.org/abs/2510.05614arxiv:2510.05614Tue, 07 Oct 2025 00:00:00 GMTBeomjun Choi, Kyeongsu Choi, Dongjun Noh — In this paper, we construct a pancake-like ancient compact solution with flat sides to the Gauss curvature flow, contained in a slab. Also, we construct sausage-like ancient compact solutions to the α-Gauss curvature flow with α>1/2, asymptotic to a round cylinder.Other curvature flowsGauss curvature flowAncient solutions https://arxiv.org/abs/2504.02804arxiv:2504.02804Thu, 03 Apr 2025 00:00:00 GMTKyeongsu Choi, Yi Lai — For an ancient Ricci flow asymptotic to a compact integrable shrinker, or a Ricci flow developing a finite-time singularity modelled on the shrinker, we establish the long-time existence of a harmonic map heat flow between the Ricci flow and the shrinker for all times. This provides a global parabolic gauge for the Ricci flow and implies the uniqueness of the tangent flow without modulo any diffeomorphisms. We present two main applications: First, we construct and classify all ancient Ricci flows asymptotic to any compact integrable shrinker, showing that they converge exponentially. Second, we obtain the optimal convergence rate at singularities modelled on the shrinker, characterized by the first negative eigenvalue of the stability operator for the entropy. In particular, we show that any Ricci flow developing a round Sⁿ singularity converges at least at the rate (−t)^((n+1)/(n-1)).Ricci flowRicci solitonsAncient solutionsSingularitiesStability & uniquenessLong-time existence & convergenceHarmonic maps & DeTurck trickEigenvalues & spectrum https://arxiv.org/abs/2412.10581arxiv:2412.10581Fri, 13 Dec 2024 00:00:00 GMTKyeongsu Choi, Robert Haslhofer — In this paper, we classify all noncollapsed singularities of the mean curvature flow in ℝ⁴. Specifically, we prove that any ancient noncollapsed solution either is one of the classical historical examples (namely ℝ^j× S^(3-j), ℝ×2d-bowl, ℝ×2d-oval, the rotationally symmetric 3d-bowl, or a cohomogeneity-one 3d-oval), or belongs to the 1-parameter family of ℤ₂× O₂-symmetric 3d-translators constructed by Hoffman-Ilmanen-Martin-White, or belongs to the 1-parameter family of ℤ₂²× O₂-symmetric ancient 3d-ovals constructed by Du-Haslhofer. In light of the five prior papers on the classification program in ℝ⁴ from our collaborations with Du, Hershkovits, and Choi-Daskalopoulos-Sesum, the major remaining challenge is the case of mixed behaviour, where the convergence to the round bubble-sheet is fast in x₁-direction, but logarithmically slow in x₂-direction. To address this, we prove a differential neck theorem, which allows us to capture the (dauntingly small) slope in x₁-direction. To establish the differential neck theorem, we introduce a slew of new ideas of independent interest, including switch and differential Merle-Zaag dynamics, anisotropic barriers, and propagation of smallness estimates. Applying our differential neck theorem, we show that every noncompact strictly convex solution is selfsimilarly translating, and also rule out exotic ovals.Mean curvature flowAncient solutionsSelf-shrinkers & translatorsSingularities(Non)collapsing3-manifoldsHomogeneous spacesBoundaries & non-compactRotational symmetry & warped products https://arxiv.org/abs/2409.01463arxiv:2409.01463Mon, 02 Sep 2024 00:00:00 GMTOtis Chodosh, Kyeongsu Choi, Christos Mantoulidis, Felix Schulze — Bamler–Kleiner recently proved a multiplicity-one theorem for mean curvature flow in R^3 and combined it with the authors' work on generic mean curvature flows to fully resolve Huisken's genericity conjecture. In this paper we show that a short density-drop theorem plus the Bamler–Kleiner multiplicity-one theorem for tangent flows at the first nongeneric singular time suffice to resolve Huisken's conjecture – without relying on the strict genus drop theorem for one-sided ancient flows previously established by the authors.Mean curvature flowAncient solutionsSingularities https://arxiv.org/abs/2405.10664arxiv:2405.10664Fri, 17 May 2024 00:00:00 GMTKyeongsu Choi, Dong-Hwi Seo, Wei-Bo Su, Kai-Wei Zhao — In this paper, we prove that an ancient smooth curve shortening flow with finite-entropy embedded in ℝ² has a unique tangent flow at infinity. To this end, we show that its rescaled flows backwardly converge to a line with multiplity m≥ 3 exponentially fast in any compact region, unless the flow is a shrinking circle, a static line, a paper clip, or a translating grim reaper. In addition, we figure out the exact numbers of tips, vertices, and inflection points of the curves at negative enough time. Moreover, the exponential growth rate of graphical radius and the convergence of vertex regions to grim reaper curves will be shown.Curve shorteningAncient solutionsSelf-shrinkers & translatorsSingularitiesStability & uniquenessLong-time existence & convergence https://arxiv.org/abs/2405.01062arxiv:2405.01062Thu, 02 May 2024 00:00:00 GMTKyeongsu Choi, Jiuzhou Huang, Taehun Lee — We construct an I-family of ancient graphical mean curvature flows over a minimal hypersurface in ℝⁿ⁺¹ of finite total curvature with the Morse index I by establishing exponentially fast convergence in terms of |x|²−t. As a corollary, we show that these ancient flows have finite total curvature and finite mass drop. Moreover, one family of these flows is mean convex by a pointwise estimate.Mean curvature flowAncient solutionsLong-time existence & convergenceMinimal & CMC surfaces https://arxiv.org/abs/2305.19137arxiv:2305.19137Tue, 30 May 2023 00:00:00 GMTKyeongsu Choi, Robert Haslhofer, Or Hershkovits — We consider the profile function of ancient ovals and of noncollapsed translators. Recall that pioneering work of Angenent-Daskalopoulos-Sesum (JDG '19, Annals '20) gives a sharp C⁰-estimate and a quadratic concavity estimate for the profile function of two-convex ancient ovals, which are crucial in their papers as well as a slew of subsequent papers on ancient solutions of mean curvature flow and Ricci flow. In this paper, we derive a sharp gradient estimate, which enhances their C⁰-estimate, and a sharp Hessian estimate, which can be viewed as converse of their quadratic concavity estimate. Motivated by our forthcoming work on ancient noncollapsed flows in ℝ⁴, we derive these estimates in the context of ancient ovals in ℝ³ and noncollapsed translators in ℝ⁴, though our methods seem to apply in other settings as well.Ricci flowMean curvature flowAncient solutionsSelf-shrinkers & translators(Non)collapsing https://arxiv.org/abs/2304.11617arxiv:2304.11617Sun, 23 Apr 2023 00:00:00 GMTKyeongsu Choi, Minhyun Kim, Taehun Lee — We establish curvature estimates for anisotropic Gauss curvature flows. By using this, we show that given a measure μ with a positive smooth density f, any solution to the Lₚ Minkowski problem in ℝⁿ⁺¹ with p ≤ −n+2 is a hypersurface of class C^(1,1). This is a sharp result because for each p∈ [−n+2,1) there exists a convex hypersurface of class C^(1,1/(n+p-1)) which is a solution to the Lₚ Minkowski problem for a positive smooth density f. In particular, the C^(1,1) regularity is optimal in the case p=−n+2 which includes the logarithmic Minkowski problem in ℝ³.Other curvature flowsGauss curvature flowAnisotropic & nonlocal flows https://arxiv.org/abs/2302.08409arxiv:2302.08409Thu, 16 Feb 2023 00:00:00 GMTOtis Chodosh, Kyeongsu Choi, Felix Schulze — We show that the mean curvature flow of a generic closed surface in ℝ³ avoids multiplicity one tangent flows that are not round spheres/cylinders. In particular, we show that any non-cylindrical self-shrinker with a cylindrical end cannot arise generically.Mean curvature flowSelf-shrinkers & translatorsSingularities