https://neckpinch.com/authors/hongwei-xu/New arXiv papers by Hongwei Xu, from Neckpinch Tue, 10 Nov 2026 04:10:28 GMT https://neckpinch.com/papers/2004.14258/arxiv:2004.14258Wed, 29 Apr 2020 00:00:00 GMTDong Pu, Jingjing Su, Hongwei Xu — In this paper, we investigate the mean curvature flow of compact surfaces in 4-dimensional space forms. We prove the convergence theorems for the mean curvature flow under certain pinching conditions involving the normal curvature, which generalise Baker-Nguyen's convergence theorem.Mean curvature flowCurvature pinchingSphere theorems https://neckpinch.com/papers/2004.13607/arxiv:2004.13607Tue, 28 Apr 2020 00:00:00 GMTLi Lei, Hong-Wei Xu — In this survey, we will focus on the mean curvature flow theory with sphere theorems, and discuss the recent developments on the convergence theorems for the mean curvature flow of arbitrary codimension inspired by the Yau rigidity theory of submanifolds. Several new differentiable sphere theorems for submanifolds are obtained as consequences of the convergence theorems for the mean curvature flow. It should be emphasized that Theorem 4.1 is an optimal convergence theorem for the mean curvature flow of arbitrary codimension, which implies the first optimal differentiable sphere theorem for submanifolds with positive Ricci curvature. Finally, we present a list of unsolved problems in this area.Mean curvature flowSphere theoremsStability & uniqueness https://neckpinch.com/papers/1910.05496/arxiv:1910.05496Sat, 12 Oct 2019 00:00:00 GMTLi Lei, Hongwei Xu, Entao Zhao — In this paper we investigate the rigidity of ancient solutions of the mean curvature flow with arbitrary codimension in space forms. We first prove that under certain sharp asymptotic pointwise curvature pinching condition the ancient solution in a sphere is either a shrinking spherical cap or a totally geodesic sphere. Then we show that under certain pointwise curvature pinching condition the ancient solution in a hyperbolic space is a family of shrinking spheres. We also obtain a rigidity result for ancient solutions in a nonnegatively curved space form under an asymptotic integral curvature pinching condition.Mean curvature flowAncient solutionsCurvature pinchingSphere theoremsStability & uniqueness https://neckpinch.com/papers/1605.07963/arxiv:1605.07963Wed, 25 May 2016 00:00:00 GMTLi Lei, Hongwei Xu — In this paper, we investigate the mean curvature flow of submanifolds of arbitrary codimension in ℂℙ^m. We prove that if the initial submanifold satisfies a pinching condition, then the mean curvature flow converges to a round point in finite time, or converges to a totally geodesic submanifold as t → ∞. Consequently, we obtain a new differentiable sphere theorem for submanifolds in ℂℙ^m. Our work improves the convergence theorem for mean curvature flow due to Pipoli and Sinestrari .Mean curvature flowSingularitiesCurvature pinchingSphere theorems