On the Convergence of Axially Symmetric Volume Preserving Mean Curvature Flow

2011/08/30 by Maria Athanassenas, Athanassenas, Maria, Sevvandi Kandanaarachchi +1
Mathematics · #53C44 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Mathematical Dynamics and Fractals #Point processes and geometric inequalities #math.DG #msc:53C44

paper · pdf · doi:10.48550/arxiv.1108.5849

12 pages, 2 figures

arxiv created 2011/08/30 · openalex publication_date 2011/08/30 · arxiv updated 2011/08/31 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28

Abstract

We study the convergence of an axially symmetric hypersurface evolving by volume preserving mean curvature flow. Assuming the surface is not pinching off along the axis at any time during the flow, and without any additional conditions, as for example on the curvature, we prove that it converges to a hemisphere, when the hypersurface has a free boundary and satisfies Neumann boundary data, and to a sphere when it is compact without boundary.

Related

• Volume preserving non homogeneous mean curvature flow in hyperbolic space
• Volume-preserving flow by powers of the m-th mean curvature
• Volume preserving mean curvature flow in the Hyperbolic space
• Volume-Preserving flow by powers of the mth mean curvature in the hyperbolic space
• Singularities of Axially Symmetric Volume Preserving Mean Curvature Flow