The inverse mean curvature flow perpendicular to the sphere

Abstract

We consider the smooth inverse mean curvature flow of strictly convex hypersurfaces with boundary embedded in $\mathbb {R}^{n+1},$ which are perpendicular to the unit sphere from the inside. We prove that the flow hypersurfaces converge to the embedding of a flat disk in the norm of $C^{1,\beta },$ $\beta <1$.

Appendix

We use this appendix to observe that in general the inverse mean curvature flow with a Neumann boundary condition may not be expected to converge globally to a minimal hypersurface as proved above. Indeed, we shall construct a counterexample for boundary manifolds arbitrarily close to the sphere.

We choose $\Sigma$ to be a rotationally symmetric ellipsoid. We will consider equations (2) replacing $\mathbb {S}^n$ with $\Sigma$ and $\tilde{N}$ with the outward normal to $\Sigma$. We start from rotationally symmetric, strictly convex initial data and flow by inverse mean curvature flow.

We firstly observe that the flow from such initial data may only exist for a finite time: Suppose not. An easy extension of Lemma 1 yields

$$\begin{aligned} H_i \tilde{n}^i = -H \check{h}_{ij}\nu ^i\nu ^j, \end{aligned}$$

where $\check{h}_{ij}$ is the second fundamental form of the ellipsoid (with respect to the outwards pointing normal vector). Due to the convexity of the ellipsoid, we may proceed as in Proposition 2 to get an upper bound on H, and so a lower bound on the speed of the flow. Since the boundary is constrained to move with a speed uniformly bounded below around the ellipsoid, the boundary must meet itself in finite time. At this point from standard calculations on rotationally symmetric surfaces, one principal curvature of the manifold must become infinite. By mean convexity we have a singularity of the flow in the sense that $\Vert A\Vert ^2$ blows up everywhere on the boundary, and the classical flow must stop. We therefore have that for all such initial data a finite time singularity occurs.

We know that if the flow does converge to a minimal surface then, since rotational symmetry is preserved by the flow, it must converge to either a catenoid or a flat plane. The former of these options is not possible since it necessitates a change of topology, and so due to the boundary condition such a global singularity may only occur at the plane of reflectional symmetry of the boundary ellipsoid.

We may now construct strictly convex, rotationally symmetric initial data as in Fig. 1 such that this initial data passes through the minimal surface—such data may always be constructed if the boundary ellipsoid is flattened in the axis of rotation. While the flow remains parabolic it may only move in one direction, and so it can never converge to this plane.

Fig. 1

An ellipsoid with strictly convex, rotationally symmetric initial data (red) that cannot converge to the only minimal surface allowed by the boundary condition (dotted) (colour figure online)

Hence, for any non-spherical rotational ellipsoid, there exist convex rotationally symmetric initial data such the the flow forms a finite time singularity and cannot converge to a (smooth) minimal surface.