https://neckpinch.com/authors/ao-sun/New arXiv papers by Ao Sun, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2609.19390arxiv:2609.19390Wed, 16 Sep 2026 00:00:00 GMTAo Sun, Jinxin Xue — In [SX25], we introduced the notion of nondegenerate cylindrical singularity using the ideas from dynamical systems. In this paper, we further study the properties of these singularities, showing that they are stable under small perturbations. Furthermore, we show that we can perturb some mean curvature flow with degenerate singularities to produce nondegenerate singularities.Mean curvature flowSingularitiesStability & uniqueness https://arxiv.org/abs/2607.23879arxiv:2607.23879Sun, 26 Jul 2026 00:00:00 GMTJingwen Chen, Tang-Kai Lee, Ao Sun, Xinrui Zhao — Given an embedded shrinker Σ in ℝⁿ⁺¹ that is either closed, asymptotically conical, or a Cartesian product of such a shrinker with ℝ^k, we construct a closed embedded mean curvature flow whose tangent flow at the first singularity is modeled on Σ. We also prescribe the first-order asymptotics of the tangent flow. This result is a consequence of a more general theorem that allows us to construct mean curvature flows with an additional force whose tangent flow and first-order asymptotics at the first singularity are prescribed.Mean curvature flowRicci solitonsSingularitiesBoundaries & non-compact https://arxiv.org/abs/2509.01707arxiv:2509.01707Mon, 01 Sep 2025 00:00:00 GMTAo Sun, Zhihan Wang, Jinxin Xue — In this paper, we study the k-cylindrical singular set of mean curvature flow in ℝⁿ⁺¹ for each 1≤ k≤ n-1. We prove that they are locally contained in a k-dimensional C^(2,α)-submanifold after removing some lower-dimensional parts. Moreover, if the k-cylindrical singular set is a k-submanifold, then its curvature is determined by the asymptotic profile of the flow at these singularities. As a byproduct, we provide a detailed asymptotic profile and graphical radius estimate at these singularities. The proof is based on a new L²-distance non-concentration property that we introduced in [SWX25], modified into a relative version that allows us to modulo those low eigenmodes that are not decaying fast enough and do not contribute to the curvature of the singular set.Mean curvature flowSingularities https://arxiv.org/abs/2501.16678arxiv:2501.16678Tue, 28 Jan 2025 00:00:00 GMTAo Sun, Zhihan Wang, Jinxin Xue — In this paper, we study the properties of nondegenerate cylindrical singularities of mean curvature flow. We prove they are isolated in spacetime and provide a complete description of the geometry and topology change of the flow passing through the singularities. Particularly, the topology change agrees with the level sets change near a critical point of a Morse function, which is the same as performing surgery. The proof is based on a new L²-distance monotonicity formula, which allows us to derive a discrete almost monotonicity of the "decay order", a discrete mean curvature flow analog to Almgren's frequency function.Mean curvature flowSingularitiesSurgery https://arxiv.org/abs/2412.03475arxiv:2412.03475Wed, 04 Dec 2024 00:00:00 GMTAlexander Mramor, Ao Sun — In this article we show that generally almost regular flows, introduced by Bamler and Kleiner, in closed 3-manifolds will either go extinct in finite time or flow to a collection of smooth embedded minimal surfaces, possibly with multiplicity. Using a perturbative argument then we construct piecewise almost regular flows which either go extinct in finite time or flow to a stable minimal surface, possibly with multiplicity. We apply these results to construct minimal surfaces in 3-manifolds in a variety of circumstances, mainly novel from the point of the view that the arguments are via parabolic methods.Mean curvature flowSingularities3-manifoldsMinimal & CMC surfaces https://arxiv.org/abs/2402.04521arxiv:2402.04521Wed, 07 Feb 2024 00:00:00 GMTJingwen Chen, Ao Sun — We construct new examples of immortal mean curvature flow of smooth embedded connected hypersurfaces in closed manifolds, which converge to minimal hypersurfaces with multiplicity 2 as time approaches infinity.Mean curvature flowLong-time existence & convergenceMinimal & CMC surfaces https://arxiv.org/abs/2312.17457arxiv:2312.17457Fri, 29 Dec 2023 00:00:00 GMTJingwen Chen, Ao Sun — We construct a new example of an immortal mean curvature flow of smooth embedded connected surfaces in ℝ³, which converges to a plane with multiplicity 2 as time approaches infinity.Mean curvature flowLong-time existence & convergence https://arxiv.org/abs/2308.05923arxiv:2308.05923Fri, 11 Aug 2023 00:00:00 GMTAdrian Chun-Pong Chu, Ao Sun — We show that for certain one-parameter families of initial conditions in ℝ³, when we run mean curvature flow, a genus one singularity must appear in one of the flows. Moreover, such a singularity is robust under perturbation of the family of initial conditions. This contrasts sharply with the case of just a single flow. As an application, we construct an embedded, genus one self-shrinker with entropy lower than a shrinking doughnut.Mean curvature flowSelf-shrinkers & translatorsSingularities https://arxiv.org/abs/2301.08224arxiv:2301.08224Thu, 19 Jan 2023 00:00:00 GMTAo Sun, Zhihan Wang — Given a smooth closed embedded self-shrinker S with index I in ℝⁿ, we construct an I-dimensional family of complete translators polynomially asymptotic to S×ℝ at infinity, which answers a long-standing question by Ilmanen. We further prove that ℝⁿ⁺¹ can be decomposed in many ways into a one-parameter family of closed sets coprod_(a∈ ℝ) Tₐ, and each closed set Tₐ contains a complete translator asymptotic to S×ℝ at infinity. If the closed set Tₐ fattens, namely it has nonempty interior, then there are at least two translators asymptotic to each other at an exponential rate, which can be viewed as a kind of nonuniqueness. We show that this fattening phenomenon is non-generic but indeed happens.Mean curvature flowSelf-shrinkers & translators https://arxiv.org/abs/2210.00419arxiv:2210.00419Sun, 02 Oct 2022 00:00:00 GMTAo Sun, Jinxin Xue — This paper studies the dynamics of mean curvature flow as it approaches a cylindrical singularity. We proved that the rescaled mean curvature flow converging to a smooth generalized cylinder can be written as a graph over the cylinder in a ball of radius K√t, and a normal form of the asymptotics. Using the normal form, we can define the nondegeneracy of cylindrical singularities, and we show that nondegenerate cylindrical singularities are isolated in space, have a mean convex neighborhood, and are type-I.Mean curvature flowSingularities https://arxiv.org/abs/2204.07896arxiv:2204.07896Sun, 17 Apr 2022 00:00:00 GMTAo Sun, Jinxin Xue — The sphere is well-known as the only generic compact shrinker for mean curvature flow (MCF). In this paper, we characterize the generic dynamics of MCFs with a spherical singularity. In terms of the level set flow formulation of MCF, we establish that generically the arrival time function of level set flow with spherical singularity has at most C² regularity.Mean curvature flowRicci solitonsSingularitiesWeak & singular flows https://arxiv.org/abs/2108.02326arxiv:2108.02326Thu, 05 Aug 2021 00:00:00 GMTAo Sun, Jonathan J. Zhu — We show that S²× S² is isolated as a shrinking Ricci soliton in the space of metrics, up to scaling and diffeomorphism. We also prove the same rigidity for S²× N, where N belongs to a certain class of closed Einstein manifolds. These results are the Ricci flow analogues of our results for Clifford-type shrinking solitons for the mean curvature flow.Ricci flowMean curvature flowRicci solitonsEinstein metricsSmale conjecture & diffeomorphismsStability & uniqueness https://arxiv.org/abs/2107.05066arxiv:2107.05066Sun, 11 Jul 2021 00:00:00 GMTAo Sun, Jinxin Xue — This is the second paper in the series to study the generic dynamics of mean curvature flows. We study the initial perturbation of mean curvature flows, whose first singularity is modeled by an asymptotically conical shrinker. The noncompactness of the limiting shrinker creates essential difficulties. We introduce the Feynman-Kac formula to get precise asymptotic behaviour of the linearized rescaled mean curvature equation along an orbit. We also develop the invariant cone method for the noncompact setting for the local dynamics near the shrinker. As a consequence, we prove that after a generic initial perturbation, the perturbed rescaled mean curvature flow avoids the conical singularity.Mean curvature flowRicci solitonsSingularitiesLimit spaces & RCDBoundaries & non-compact