https://neckpinch.com/authors/zhihan-wang/New arXiv papers by Zhihan Wang, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT 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/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