[Paper Review] On the Convergence of Axially Symmetric Volume Preserving Mean Curvature Flow

Maria Athanassenas, Sevvandi Kandanaarachchi|arXiv (Cornell University)|Aug 30, 2011

Geometric Analysis and Curvature Flows4 references4 citations

TL;DR

This paper establishes the long-time existence and convergence of axially symmetric volume-preserving mean curvature flow for hypersurfaces without pinching along the axis. Under mild geometric assumptions—no curvature bounds and only a lower height bound—it proves convergence to a hemisphere for surfaces with Neumann boundary conditions and to a sphere for compact, boundaryless surfaces, leveraging axial symmetry and gradient estimates to avoid curvature constraints.

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.

Motivation & Objective

• To analyze the long-time behavior of axially symmetric hypersurfaces evolving under volume-preserving mean curvature flow.
• To establish convergence to a surface of constant mean curvature under minimal geometric assumptions.
• To remove the need for curvature bounds or convexity assumptions in the convergence analysis.
• To extend prior results on spherical convergence to the case of free boundary Neumann conditions.
• To prove convergence to a sphere for compact, boundaryless axially symmetric surfaces using similar techniques.

Proposed method

• Utilizes axial symmetry to reduce the problem to a one-dimensional flow of the generating curve.
• Implements a decomposition of the surface into a cylindrical part $ R_t^eta $ and caps $ C_t^eta $ based on normal inclination.
• Applies height estimates via comparison with cylinders and balls to bound the total length and surface area.
• Derives gradient estimates using $ v = \langle \nu, \omega \rangle^{-1} $ and $ \tilde{v} = \langle \nu, \mathbf{i}_1 \rangle^{-1} $, controlling the normal angle.
• Establishes uniform bounds on the mean curvature average $ h(t) $ via integration by parts and vanishing boundary terms.
• Uses long-time existence from curvature estimates and gradient control, leading to convergence to a constant mean curvature surface.

Experimental results

Research questions

• RQ1 Can axially symmetric volume-preserving mean curvature flow converge to a sphere or hemisphere without curvature or convexity assumptions?
• RQ2 What geometric conditions ensure long-time existence and convergence for such flows?
• RQ3 How can height and gradient estimates be used to control the evolution without curvature bounds?
• RQ4 Does the Neumann boundary condition at the supporting plane lead to convergence to a hemisphere?
• RQ5 Can the same method be adapted to compact, boundaryless surfaces to prove spherical convergence?

Key findings

• The flow exists for all time, i.e., $ T_{\max} = \infty $, under the assumption of no pinching along the axis.
• For surfaces with Neumann boundary conditions on a plane, the flow converges to a hemisphere.
• For compact, boundaryless axially symmetric surfaces, the flow converges to a sphere.
• The convergence is guaranteed without any curvature bounds or convexity assumptions.
• Gradient estimates $ v \leq c $ on the cylindrical part and $ \tilde{v} \leq \alpha $ on the caps are uniformly controlled.
• The mean curvature average $ h(t) $ remains uniformly bounded, ensuring convergence to a constant mean curvature surface.