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
([38] and
[41]) of at least three joint papers with
**Ben Chow.** For
example,
in
[41], 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.

[38].
Aleksandrov Reflection and Nonlinear Evolution Equations, I: The
n-Sphere and n-Ball (with Bennett Chow), |