  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.
