https://neckpinch.com/authors/junyoung-park/New arXiv papers by Junyoung Park, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2608.11011arxiv:2608.11011Tue, 11 Aug 2026 00:00:00 GMTJunyoung Park — In this paper, we prove that a smooth, properly embedded ancient mean curvature flow that is asymptotic to an O(n)× O(m) symmetric minimal quadratic cone for n +m ≥ 10, and lies on one side of the cone has to have unique asymptotics in the parabolic region. When additionally assuming mean convexity, we upgrade unique asymptotics to full uniqueness, and show that such flow has to be a stationary flow given by one of the leaves of the Hardt-Simon foliation. This is the first rigidity/unique asymptotics theory for ancient mean curvature flow with a singular minimal cone as the asymptotic model.Mean curvature flowAncient solutionsStability & uniquenessMinimal & CMC surfaces https://arxiv.org/abs/2601.22368arxiv:2601.22368Thu, 29 Jan 2026 00:00:00 GMTJunyoung Park — In the first part of the paper, we prove the existence of longtime solution to mean curvature flow starting from a graph of a continuous function defined over a slab. Then, we establish dynamical stability results for various types of graphical translators to mean curvature flow, namely the grim reaper, two dimensional graphical translators, and asymptotically cylindrical translators.Mean curvature flowSelf-shrinkers & translatorsSurfacesStability & uniquenessLong-time existence & convergence https://arxiv.org/abs/2502.07210arxiv:2502.07210Tue, 11 Feb 2025 00:00:00 GMTJunyoung Park — The goal of this paper is to relax convexity assumption on some classical results in mean curvature flow. In the first half of the paper, we prove a generalized version of Hamilton's differential Harnack inequality which holds for mean convex solutions to mean curvature flow with a lower bound on (λ₁)/H where λ₁ is the smallest principal curvature. Then, we use classical maximum principle to provide several characterizations of family of shrinking spheres for closed, mean convex ancient solution to mean curvature flow with a lower bound on (λ₁ + .. + λₖ)/H for some 1 ≤ k ≤ d-1, where λ₁ ≤ λ₂ ≤ .. ≤ λ_d are the principal curvatures.Mean curvature flowAncient solutionsHarnack inequalities