https://neckpinch.com/authors/xiao-liu-wang/New arXiv papers by Xiao-Liu Wang, from Neckpinch Tue, 03 Nov 2026 08:41:12 GMT https://neckpinch.com/papers/2610.35064/arxiv:2610.35064Mon, 26 Oct 2026 00:00:00 GMTZe-Yu Ye, Xiao-Liu Wang — In this paper, we study the singularity behavior of immersed locally convex closed plane curves evolving under generalized anisotropic Gage area-preserving flows with slow curvature speeds of the form κ_σ^α, where 0<α≤ 1. Here κ_σ denotes the anisotropic curvature associated with a prescribed anisotropy function σ. We prove that every finite-time blow-up is of Type-II provided that the Euclidean length of the evolving curve remains uniformly bounded away from zero. This result holds without any symmetry assumptions on either the initial curve or the anisotropy. When the initial signed area is nonzero, this length lower bound is automatically satisfied, and thus any finite-time singularity is necessarily of Type-II. By means of Hamilton's rescaling method combined with a differential Harnack estimate, we show that along a suitable blow-up subsequence, the rescaled curves at the selected times converge locally to the profile of a translating solution of the corresponding anisotropic curvature flow V=κ_σ^α. We also establish sufficient conditions for finite-time singularity formation in terms of the initial signed area and anisotropic length. Moreover, we construct locally convex initial curves arbitrarily close to the Wulff curve for which a finite-time Type-II singularity occurs.Other curvature flowsVolume-preserving & constrained flowsAnisotropic & nonlocal flowsSelf-shrinkers & translatorsSingularitiesHarnack inequalities https://neckpinch.com/papers/2302.07831/arxiv:2302.07831Fri, 03 Feb 2023 00:00:00 GMTXinfu Chen, Bendong Lou, Xiaoliu Wang, Lixia Yuan — We consider a graphical mean curvature flow in a cylinder with Robin boundary conditions, which arises as a geometric model for interface motion in the singular limit of the Allen–Cahn equation with nonlinear boundary conditions. It was shown in [26] that, in the planar case, every solution converges to a translating Grim Reaper with a fixed profile and finite speed. In this paper, we investigate the radially symmetric problem in higher dimensions and reveal a completely different asymptotic dynamics caused by the spatial dimension. In contrast to the planar case, there is no fixed translating profile governing the long-time behaviour. Instead, the solution propagates with an exponentially increasing speed, while both the gradient |Du| (away from the center) and the instantaneous speed uₜ diverge exponentially as t→∞. This reveals a fundamentally different asymptotic behaviour induced by the interaction between the Robin boundary condition and the spatial dimension, that is, the translating profile continuously degenerates and becomes asymptotically ray-like. Since the equation becomes asymptotically degenerate and no uniform-in-time C⁰, C¹, or C² estimates are available, our analysis relies on a new approach based on the zero number argument.Mean curvature flowSelf-shrinkers & translatorsBoundaries & non-compactLong-time existence & convergenceRotational symmetry & warped productsPhase fields & multiphase flows