[Paper Review] Mean curvature flow with free boundary outside a hypersphere

Glen Wheeler, Valentina‐Mira Wheeler|arXiv (Cornell University)|May 30, 2014

Geometric Analysis and Curvature Flows9 references4 citations

TL;DR

This paper investigates the mean curvature flow of a topological annulus with free boundary on the exterior of an $n$-sphere and Dirichlet boundary on a translated $n$-sphere of radius $R>1$. Using rotational and reflective symmetry, along with Killing vector fields to preserve graphicality, the authors prove global existence and convergence to a minimal surface for initially mean concave/convex graphical data, establishing long-time behavior and curvature bounds under geometric constraints.

ABSTRACT

The purpose of this paper is twofold: firstly, to establish sufficient conditions under which the mean curvature flow supported on a hypersphere with exterior Dirichlet boundary exists globally in time and converges to a minimal surface, and secondly, to illustrate the application of Killing vector fields in the preservation of graphicality for the mean curvature flow with free boundary. To this end we focus on the mean curvature flow of a topological annulus with inner boundary meeting a standard n-sphere in \R^{n+1} perpendicularly and outer boundary fixed to an (n-1)-sphere with radius R>1 at a fixed height h. We call this the \emph{sphere problem}. Our work is set in the context of graphical mean curvature flow with either symmetry or mean concavity/convexity restrictions. For rotationally symmetric initial data we obtain, depending on the exact configuration of the initial graph, either long time existence and convergence to a minimal hypersurface with boundary or the development of a finite-time curvature singularity. With reflectively symmetric initial data we are able to use Killing vector fields to preserve graphicality of the flow and uniformly bound the mean curvature pointwise along the flow. Finally we prove that the mean curvature flow of an initially mean concave/convex graphical surface exists globally in time and converges to a piece of a minimal surface.

Motivation & Objective

• To establish sufficient conditions for global existence and convergence to a minimal surface in mean curvature flow with free boundary outside a hypersphere.
• To analyze the behavior of mean curvature flow on a topological annulus where the inner boundary meets a standard $n$-sphere perpendicularly and the outer boundary is fixed at a translated $n$-sphere of radius $R>1$.
• To demonstrate the utility of Killing vector fields in preserving graphicality and uniformly bounding mean curvature during the flow.
• To extend global existence results to initially mean concave or convex graphical surfaces under symmetry assumptions.
• To complement existing interior sphere results by studying the exterior problem, providing a dual perspective on free boundary mean curvature flow.

Proposed method

• Formulate the problem as a graphical mean curvature flow on a manifold with Neumann (free boundary) and Dirichlet (fixed height) boundaries.
• Use rotational symmetry to reduce the problem to a radial setting, enabling analysis of curvature evolution and existence time.
• Apply reflectively symmetric initial data to exploit Killing vector fields that preserve graphicality and control gradient growth.
• Employ barrier constructions and the Hopf lemma to derive a priori lower bounds on the normal component $s = \langle \nu^{M_t}, e_{n+1} \rangle$.
• Use the monotonicity of the area functional and uniform bounds on derivatives to prove convergence to a minimal hypersurface.
• Leverage comparison principles and local Fermi coordinates to control boundary curvature and prevent singularities.

Experimental results

Research questions

• RQ1 Under what geometric and initial data conditions does the mean curvature flow with free boundary outside a hypersphere exist globally in time?
• RQ2 How can Killing vector fields be used to preserve graphicality and control mean curvature in the presence of free boundary conditions?
• RQ3 What is the long-term behavior of the mean curvature flow when the initial surface is mean concave or convex and symmetric?
• RQ4 Can the exterior problem on a sphere be analyzed in duality with the interior problem studied by Stahl?
• RQ5 What conditions ensure that the flow converges to a minimal hypersurface with boundary?

Key findings

• For rotationally symmetric initial data, the mean curvature flow either exists globally and converges to a minimal hypersurface or develops a finite-time curvature singularity, depending on the initial configuration.
• With reflectively symmetric initial data, the use of Killing vector fields ensures graphicality is preserved and the mean curvature is uniformly bounded along the flow.
• The flow of an initially mean concave or convex graphical surface exists globally in time and converges to a minimal hypersurface, provided the initial data satisfy the required symmetry and graphicality conditions.
• A priori lower bounds on $s = \langle \nu^{M_t}, e_{n+1} \rangle$ are established via the Hopf lemma and barrier arguments, preventing degeneracy at the Neumann boundary.
• The global existence and convergence result is proven using the area functional's monotonicity and uniform bounds on all derivatives, implying $M_t \to M_\infty$ with $H \equiv 0$.
• The initial condition $\langle F_0, e_{n+1} \rangle > 0$ and the barrier from the flat plane at zero height ensure that $s$ remains bounded below, preserving the flow's regularity.