Mean curvature flow of surfaces in a hyperkähler 4-manifold

2019/02/02 by Hongbing Qiu, Qiu, Hongbing, Linlin Sun +1
Computer Science · Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Topological and Geometric Data Analysis #math.DG

paper · pdf · doi:10.48550/arxiv.1902.00645

openalex publication_date 2019/02/02 · arxiv created 2020/11/24 · arxiv updated 2020/11/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we firstly prove that every hyper-Lagrangian submanifold L²ⁿ (n > 1) in a hyperkähler 4n-manifold is a complex Lagrangian submanifold. Secondly, we demonstrate an optimal rigidity theorem with the condition on the complex phase map of self-shrinking surfaces in ℝ⁴. Last but not least, by using the previous rigidity result, we show that the mean curvature flow from a closed surface with the image of the complex phase map contained in \mathbbS²∖\mathbbS¹₊ in a hyperkähler 4-manifold does not develop any Type \Rmn1 singularity.

Citations

• Convexity estimates for mean curvature flow and singularities of mean convex surfaces

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