  This work considers the question of whether mean-curvature flow can be
modified to avoid the formation of singularities. We analyze the
finite-elements discretization and demonstrate why the original flow can result
in numerical instability due to division by zero. We propose a variation on the
flow that removes the numerical instability in the discretization and show that
this modification results in a simpler formulation for both the discretized and
continuous formulations. We discuss the properties of the modified flow and
present empirical evidence that not only does it define a stable surface
evolution for genus-zero surfaces, but that the evolution converges to a
conformal parameterization of the surface onto the sphere.
