https://neckpinch.com/authors/beomjun-choi/New arXiv papers by Beomjun Choi, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2610.10461arxiv:2610.10461Wed, 07 Oct 2026 00:00:00 GMTBeomjun Choi, Wenkui Du, Seung Chul Park, Junseo Youn — In this paper, we establish a sharp dynamical isoperimetric principle for weak mean curvature flow. We prove that, among weak mean curvature flows with fixed area of initial smooth closed hypersurfaces, the standard smoothly shrinking spherical mean curvature flow uniquely maximizes the extinction time. More precisely, for the level set flow Kₜ starting from the boundary of a smooth bounded domain Ω⊂ℝⁿ⁺¹, as well as for the integral Brakke flow \μₜ\ with initial Radon measure μ₀=𝓗ⁿllcorner∂Ω, we establish the corresponding optimal extinction time estimates \beginequation* T_{\rm ext}(Ω),\, T^B_{\rm ext}(Ω) \leq \frac12n \left(\fracP( Ω){|{\mathbbS^n}|}\right)^{\frac2n}, \endequation* where P(Ω) is the perimeter of Ω representing the area of ∂Ω, and the equality holds if and only if Ω is a round ball and the flow is the standard multiplicity-one smoothly self-shrinking round sphere. In particular, we obtain the sharp L^p-estimates for the arrival time function of a smooth bounded mean convex domain. In addition, we also establish the sharp isoperimetric inequality for the parabolic measure of the space-time track filling X of outward minimizing level set flow starting from the boundary of a smooth bounded domain Ω⊂ ℝⁿ⁺¹: \beginequation* \mathcal H_par^n+2(X) \leq \fracπ {2n(n+2)^2|{\mathbbS^n}|^{{\frac2n}}} P(Ω)^{\fracn+2n}, \endequation* where the equality holds if and only if Ω is a round ball and the flow is standard smoothly shrinking round sphere.Mean curvature flowSelf-shrinkers & translatorsWeak & singular flowsGeometric inequalities https://arxiv.org/abs/2601.09441arxiv:2601.09441Wed, 14 Jan 2026 00:00:00 GMTBeomjun Choi, Wenkui Du, Ziyi Zhao — We study compact non-selfsimilar ancient noncollapsed solutions to the mean curvature flow in ℝⁿ⁺¹, called ancient ovals. Our main result is the classification of k-ovals: any k-oval (characterized by having cylindrical blow down ℝ^k× S^(n-k) and the quadratic bending asymptotics) belongs, up to space-time rigid motions and parabolic dilations, to the family of ancient ovals constructed by Haslhofer and the second author. Assuming the nonexistence of exotic ovals (recently proved by Bamler-Lai), this yields a classification of all ancient ovals and identifies the moduli space, modulo symmetries, with an open (k-1)-simplex modulo the symmetry of simplex. Although these conclusions are contained in the recent breakthrough of Bamler-Lai classifying all ancient asymptotically cylindrical flows and resolving the mean convex neighborhood conjecture, we give an alternative argument for the independently obtained classification of k-ovals in arbitrary dimensions based on a different spectral parametrization.Mean curvature flowAncient solutions(Non)collapsingEigenvalues & spectrum 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/2504.09741arxiv:2504.09741Sun, 13 Apr 2025 00:00:00 GMTBeomjun Choi, Wenkui Du, Jingze Zhu — In this paper, we consider the classification of compact ancient noncollapsed mean curvature flows of hypersurfaces in arbitrary dimensions. More precisely, we study k-ovals in ℝⁿ⁺¹, defined as ancient noncollapsed solutions whose tangent flow at -∞ is given by ℝ^k × S^(n-k)((2(n-k)|t|)^(1/2)) for some k ∈ \1,…,n-1\, and whose fine cylindrical matrix has full rank. A significant advance achieved recently by Choi and Haslhofer suggests that the shrinking n-sphere and k-ovals together account for all compact ancient noncollapsed solutions in ℝⁿ⁺¹. We prove that k-ovals are ℤ^(k)₂ × O(n+1-k)-symmetric and are uniquely determined by (k-1)-dimensional spectral ratio parameters. This result is sharp in view of the (k-1)-parameter family of ℤ^(k)₂ × O(n+1-k)-symmetric ancient ovals constructed by Du and Haslhofer, as well as the conjecture of Angenent, Daskalopoulos and Sesum concerning the moduli space of ancient solutions. We also establish a new spectral stability theorem, which suggests the local (k-1)-rectifiability of the moduli space of k-ovals modulo space-time rigid motion and parabolic rescaling. In contrast to the case of 2-ovals in ℝ⁴, resolved by Choi, Daskalopoulos, Du, Haslhofer and Sesum, the general case for arbitrary k and n presents new challenges beyond increased algebraic complexity. In particular, the quadratic concavity estimates in the collar region and the absence of a global parametrization with regularity information pose major obstacles. To address these difficulties, we introduce a novel test tensor that produces essential gradient terms for the tensor maximum principle, and we derive a local Lipschitz continuity result by parameterizing k-ovals with nearly matching spectral ratio parameters.Mean curvature flowAncient solutionsSingularities(Non)collapsingStability & uniquenessEigenvalues & spectrum https://arxiv.org/abs/2209.04931arxiv:2209.04931Sun, 11 Sep 2022 00:00:00 GMTBeomjun Choi, Panagiota Daskalopoulos, Wenkui Du, Robert Haslhofer, Natasa Sesum — In this paper, we prove that any bubble-sheet oval for the mean curvature flow in ℝ⁴, up to scaling and rigid motion, either is the O(2)× O(2)-symmetric ancient oval constructed by Hershkovits and the fourth author, or belongs to the one-parameter family of ℤ₂²× O(2)-symmetric ancient ovals constructed by the third and fourth author. In particular, this seems to be the first instance of a classification result for geometric flows that are neither cohomogeneity-one nor selfsimilar.Mean curvature flowAncient solutionsHomogeneous spaces https://arxiv.org/abs/2109.13875arxiv:2109.13875Tue, 28 Sep 2021 00:00:00 GMTBeomjun Choi, Robert Haslhofer — We prove that Perelman's ancient ovals occur as blowup limit in 3d Ricci flow through singularities if and only if there is an accumulation of spherical singularities.Ricci flowAncient solutionsSingularities3-manifoldsWeak & singular flows https://arxiv.org/abs/2108.02944arxiv:2108.02944Fri, 06 Aug 2021 00:00:00 GMTBeomjun Choi, Robert Haslhofer — In this paper we reconcile several different approaches to Ricci flow through singularities that have been proposed over the last few years by Kleiner-Lott, Haslhofer-Naber and Bamler. Specifically, we prove that every noncollapsed limit of Ricci flows, as provided by Bamler's precompactness theorem, as well as every singular Ricci flow from Kleiner-Lott, is a weak solution in the sense of Haslhofer-Naber. We also generalize all path-space estimates from Haslhofer-Naber to the setting of noncollapsed Ricci limit flows. The key step to establish these results is a new hitting estimate for Brownian motion. A fundamental difficulty, in stark contrast to all prior hitting estimates in the literature, is the lack of lower heat kernel bounds under Ricci flow. To overcome this, we introduce a novel approach to hitting estimates that compensates for the lack of lower heat kernel bounds by making use of the heat kernel geometry of space-time.Ricci flowSingularities(Non)collapsingHeat kernelWeak & singular flowsLimit spaces & RCD