The Aleksandrov Reflection Method for Hypersurfaces Expanding by a Function of Curvatures

Let be a compact embedded hypersurface

in

which moves with speed determined at each point by a function

of its principal curvatures,

for We assume the problem is degenerat

e parabolic,

that is, that is nondecreasing in each

of the

principal curvatures We shall show

that for t > 0 the hypersurface satisfies local

a priori Lipschitz bounds outside of a convex set determined

by and lying inside its convex hull. A

s a consequence,

if expands to reach points further and

further away from

then

LE ALT="tex2html_wrap_inline15" SRC="gulliver191099/img1.gif"> converges to a round sphere after

rescaling. This is proved without estimates on curvature, in contrast

to earlier proofs of asymptotic roundness, which typically require

strict parabolicity. Our method is the parabolic analogue of

Aleksandrov's method of moving planes.

Aleksandrov's reflection method is also extended to treat generalized

solutions of this evolution problem, that is, level sets of viscosity

solutions to the corresponding geometric PDE. These generalized

solutions have recently been shown in certain cases to develop a

nonempty interior after the evolving hypersurface collides with

itself or develops singularities. We shall prove that the same

local Lipschitz bounds as in the embedded-hypersurface case hold

for the inner and outer boundaries of such a "fattened" level set

As an application, we give some new results about 1/H flow for

nonconvex hypersurfaces, which was recently investigated by Huisken

and Ilmanen. Our solutions are limits of viscosity solutions, and

are therefore solutions of a local problem, in only.

In contrast, the evolving hypersurfaces given by Huisken and Ilmanen,

which were used to prove a version of the Penrose conjecture, are

solutions of a non-local problem, valid in general asymptotically

flat Riemannian manifolds.

This is joint work with Bennett Chow.