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

  • Robert Gulliver (Minneapolis)
A3 01 (Sophus-Lie room)

Abstract

Let tex2html_wrap_inline15 be a compact embedded hypersurface

in tex2html_wrap_inline17

which moves with speed determined at each point by a function

tex2html_wrap_inline19 of its principal curvatures,

for tex2html_wrap_inline21 We assume the problem is degenerat

e parabolic,

that is, that tex2html_wrap_inline23 is nondecreasing in each

of the

principal curvatures tex2html_wrap_inline25 We shall show

that for t > 0 the hypersurface tex2html_wrap_inline15 satisfies local

a priori Lipschitz bounds outside of a convex set determined

by tex2html_wrap_inline31 and lying inside its convex hull. A

s a consequence,

if tex2html_wrap_inline15 expands to reach points further and

further away from

tex2html_wrap_inline35 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

tex2html_wrap_inline39

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 tex2html_wrap_inline17 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.

seminar
4/18/24 5/30/24

Oberseminar Analysis

Universität Leipzig Augusteum - A314

Anne Dornfeld

MPI for Mathematics in the Sciences Contact via Mail

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