https://neckpinch.com/authors/jinxin-xue/New arXiv papers by Jinxin Xue, 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/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/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/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