https://neckpinch.com/authors/yiqi-huang/New arXiv papers by Yiqi Huang, 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/2510.23499arxiv:2510.23499Mon, 27 Oct 2025 00:00:00 GMTYiqi Huang, Xinrui Zhao — We prove that if a rescaled mean curvature flow is a global graph over the round cylinder with small gradient and converges super-exponentially fast, then it must coincide with the cylinder itself. We also show that the result is sharp with counter-examples of local graphs at arbitrarily super-exponential convergence rate with the domain expanding arbitrarily fast. The first part provides the first unique continuation result in the cylindrical setting, the generic singularity model in mean curvature flow. In sharp contrast, in the second part we construct infinite-dimensional families of Tikhonov-type examples for nonlinear equations, including the rescaled mean curvature flow, showing that unique continuation fails for local graphical solutions. These examples demonstrate the essential role of global graphical assumptions in rigidity and highlight new phenomena absent in the compact case. We also construct non-product mean curvature flows that develop singular sets as prescribed lower dimensional Euclidean space at arbitrary super-exponential rates. Our construction works in great generality for a large class of non-linear equations.Mean curvature flowSingularitiesStability & uniquenessLong-time existence & convergence https://arxiv.org/abs/2510.17060arxiv:2510.17060Mon, 20 Oct 2025 00:00:00 GMTYiqi Huang, Wenshuai Jiang — We show that the intrinsic diameter of mean curvature flow in ℝ³ is uniformly bounded as one approaches the first singular time T. This confirms the bounded diameter conjecture of Haslhofer. In addition, we establish several sharp quantitative estimates: the second fundamental form A has uniformly bounded L¹-norm on each time slice, A belongs to the weak L³ space on the space-time region, and the singular set 𝓢 has finite 𝓗¹-Hausdorff measure. All of the results are optimal due to the marriage ring example and our results do not require any convexity assumptions on the surfaces. Furthermore, our arguments extend naturally to flows through singularities, yielding the same sharp estimates.Mean curvature flowSingularitiesWeak & singular flows