I am also intensely interested in applying analogues of the Aleksandrov moving-plane method to parabolic problems in geometry and PDE's. This has led to the first two ( and ) of at least three joint papers with Ben Chow. For example, in , Postscript version or PDF version we prove that a hypersurface of Euclidean (n + 1) - space which expands by an arbitrary function of its principal curvatures and eventually includes points arbitrarily far away from the origin, must become asymptotically round; provided only that the problem is parabolic or degenerate parabolic. However, it may happen that the hypersurface becomes a viscosity solution with nonempty interior; in this case, asymptotic roundness means Lipschitz-closeness to a round annulus, after rescaling to unit radius.
Aleksandrov Reflection and Nonlinear Evolution Equations, I: The
n-Sphere and n-Ball (with Bennett Chow), Calculus of
Variations and Partial Differential Equations 4,
Postscript version or