https://neckpinch.com/authors/wenkui-du/New arXiv papers by Wenkui Du, 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/2608.26619arxiv:2608.26619Thu, 27 Aug 2026 00:00:00 GMTKyeongho Bang, Eric Chen, Wenkui Du — We establish the existence and uniqueness modulo time-independent diffeomorphisms of the positively curved ancient Ricci flow (M², ∂M², g(t)) on a two-dimensional surface with boundary, assuming uniformly bounded diameter and constant positive boundary geodesic curvature. In particular, this ancient Ricci flow is rotationally symmetric, its backward limit is the flat disk, and its forward limit is a half-spherical singularity. To our knowledge, this result is the first instance of a classification result for ancient Ricci flows with boundary.Ricci flowAncient solutionsSingularitiesSurfacesStability & uniquenessRotational symmetry & warped products https://arxiv.org/abs/2607.03383arxiv:2607.03383Fri, 03 Jul 2026 00:00:00 GMTPanagiota Daskalopoulos, Wenkui Du, Natasa Sesum, Ziyi Zhao — We obtain the unique asymptotics of SO(k)× SO(n-k+1)-invariant, compact, simply-connected, factorwisely non-self-similar n-dimensional κ-solutions of the Ricci flow (Mⁿ, g(t)), where n≥ 4 and 2≤ k≤ n-2. More precisely, these κ-solutions are either ancient ovals of the Ricci flow that are diffeomorphic to the standard sphere Sⁿ, having a positive curvature operator metric g(t) and a cylindrical tangent flow at -∞, or they are a Riemannian product of a Perelman's ancient oval and a shrinking round sphere. The metric g(t) of every SO(k)× SO(n-k+1)-invariant ancient oval is represented in the form g(t)=dz⊗ dz + F²(z,t) g_(S^(k-1)) + G²(z,t)g_(S^(n-k)) (up to flipping k-1 and n-k). We obtain results about the blowdown limits of such solutions, establish the unique sharp asymptotics of the profile function G(z, t), and prove that the uniqueness of G(z, t) implies the uniqueness of F(z, t). In particular, this provides the first instance of a classification result for geometric flows represented by a coupled PDE system, opening new avenues for studying the classification of higher-dimensional κ-solutions of the Ricci flow.Ricci flowAncient solutionsSingularitiesSphere theoremsStability & uniqueness 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/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/2211.02595arxiv:2211.02595Fri, 04 Nov 2022 00:00:00 GMTWenkui Du, Jingze Zhu — We study ancient mean curvature flows in ℝⁿ⁺¹ whose tangent flow at -∞ is a shrinking cylinder ℝ^(k)× S^(n-k)(√(2(n-k)|t|)), where 1≤ k≤ n-1. We prove that the cylindrical profile function u of these flows have the asymptotics u(y,ω,τ)= (y^top Qy −2tr(Q))/|τ| + o(|τ|⁻¹) as τ→ -∞, where the cylindrical matrix Q is a constant symmetric k× k matrix whose eigenvalues are quantized to be either 0 or −frac√(2(n-k))4. Compared with the bubble-sheet quantization theorem in ℝ⁴ obtained by Haslhofer and the first author, this theorem has full generality in the sense of removing noncollapsing condition and being valid for all dimensions. In addition, we establish symmetry improvement theorem which generalizes the corresponding results of Brendle-Choi and the second author to all dimensions. Finally, we give some geometric applications of the two theorems. In particular, we obtain the asymptotics, compactness and O(n-k+1) symmetry of k-ovals in ℝⁿ⁺¹ which are ancient noncollapsed flows in ℝⁿ⁺¹ satisfying full rank condition that rk(Q)=k, and we also obtain the classification of ancient noncollapsed flows in ℝⁿ⁺¹ satisfying vanishing rank condition that rk(Q)=0.Mean curvature flowAncient solutionsSingularities(Non)collapsingLimit spaces & RCDEigenvalues & 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/2208.14280arxiv:2208.14280Tue, 30 Aug 2022 00:00:00 GMTWenkui Du, Robert Haslhofer — Some worrisome potential singularity models for the mean curvature flow are rotating ancient flows, i.e. ancient flows whose tangent flow at -∞ is a cylinder ℝ^k× S^(n-k) and that are rotating within the ℝ^k-factor. We note that while the ℝ^k-factor, i.e. the axis of the cylinder, is unique by the fundamental work of Colding-Minicozzi, the uniqueness of tangent flows by itself does not provide any information about rotations within the ℝ^k-factor. In the present paper, we rule out rotating ancient flows among all ancient noncollapsed flows in ℝ⁴.Mean curvature flowAncient solutionsSingularities(Non)collapsingStability & uniqueness https://arxiv.org/abs/2107.04443arxiv:2107.04443Fri, 09 Jul 2021 00:00:00 GMTWenkui Du, Robert Haslhofer — We consider ancient noncollapsed mean curvature flows in ℝ⁴ whose tangent flow at -∞ is a bubble-sheet. We carry out a fine spectral analysis for the bubble-sheet function u that measures the deviation of the renormalized flow from the round cylinder ℝ² × S¹(√2) and prove that for τ→ -∞ we have the fine asymptotics u(y,θ,τ)= (y^top Qy −2tr(Q))/|τ| + o(|τ|⁻¹), where Q=Q(τ) is a symmetric 2× 2-matrix whose eigenvalues are quantized to be either 0 or −1/√8. This naturally breaks up the classification problem for general ancient noncollapsed flows in ℝ⁴ into three cases depending on the rank of Q. In the case rk(Q)=0, generalizing a prior result of Choi, Hershkovits and the second author, we prove that the flow is either a round shrinking cylinder or ℝ×2d-bowl. In the case rk(Q)=1, under the additional assumption that the flow either splits off a line or is selfsimilarly translating, as a consequence of recent work by Angenent, Brendle, Choi, Daskalopoulos, Hershkovits, Sesum and the second author we show that the flow must be ℝ×2d-oval or belongs to the one-parameter family of 3d oval-bowls constructed by Hoffman-Ilmanen-Martin-White, respectively. Finally, in the case rk(Q)=2 we show that the flow is compact and SO(2)-symmetric and for τ→-∞ has the same sharp asymptotics as the O(2)×O(2)-symmetric ancient ovals constructed by Hershkovits and the second author. The full classification problem will be addressed in subsequent papers based on the results of the present paper.Mean curvature flowAncient solutionsSingularities(Non)collapsing3-manifoldsEigenvalues & spectrum