https://neckpinch.com/authors/jingze-zhu/New arXiv papers by Jingze Zhu, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2609.31610arxiv:2609.31610Fri, 25 Sep 2026 00:00:00 GMTYiqi Huang, Jingze Zhu — For a convex mean curvature flow in ℝⁿ⁺¹, the corresponding arrival time function solves a degenerate elliptic equation that becomes singular at extinction. We prove that it is C^(24,α) for any 0<α<1 in the plane, with a logarithmic modulus for its twenty-fourth derivatives. We prove that this bound is sharp combining regularity obstruction in our companion paper. We obtain asymptotic expansions of the rescaled flow to any prescribed order, including derivative estimates for the remainder. Combined with the work of Sesum for n≥ 2, this also gives the sharp C^(2,2/n) regularity for convex arrival time functions in higher dimensions.Mean curvature flow https://arxiv.org/abs/2609.31599arxiv:2609.31599Fri, 25 Sep 2026 00:00:00 GMTYiqi Huang, Jingze Zhu — We study the regularity of the arrival time function associated with mean curvature flow, formulated as a degenerate elliptic equation encoding the singular structure of the flow. We identify a systematic mechanism obstructing higher regularity, arising from higher-order asymptotic expansions near spherical singularities. We construct uncountably many low-regularity arrival time functions in all dimensions. In particular, we settle the question of smoothness in the planar case by showing that even for convex curve shortening flow the arrival time need not be C²⁵. Our companion paper proves that every such arrival time is C^(24,α) for any 0<α<1. Hence the obstruction at order 25 gives the optimal regularity threshold. Our approach introduces new analytic ingredients, including a precise correspondence between elliptic asymptotic expansions and parabolic long-time asymptotics of the associated rescaled mean curvature flow, a complexification of the arrival time equation, and a method for prescribing higher-order asymptotic expansions.Mean curvature flowCurve shorteningSingularities https://arxiv.org/abs/2504.17922arxiv:2504.17922Thu, 24 Apr 2025 00:00:00 GMTTang-Kai Lee, Keaton Naff, Jingze Zhu — We prove a parabolically scale-invariant variation of the planarity estimate in for higher codimension mean curvature flow, borrowing ideas from work of Brendle–Huisken–Sinestrari. 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.Mean curvature flowRicci solitonsAncient 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/2405.19064arxiv:2405.19064Wed, 29 May 2024 00:00:00 GMTTang-Kai Lee, Jingze Zhu — 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 ℝⁿ⁺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 C². The results also apply to mean curvature flows starting from two-spheres or generic closed surfaces.Mean curvature flowSingularitiesStability & uniqueness 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/2108.11852arxiv:2108.11852Thu, 26 Aug 2021 00:00:00 GMTJingze Zhu — We prove that smooth convex α-noncollapsed ancient mean curvature flow satisfies a quantitative curvature estimate H(y,t)≤ CH(x,t)(H(x,t)|x-y|+1)² for any pair of x,y. In other words, the rescaled curvature grows at most quadratically in terms of the rescaled extrinsic distance.Mean curvature flowAncient solutions(Non)collapsing