Inverse Mean Curvature Flow (IMCF)
The motion of surfaces by their inverse mean curvature has been studied
recently by Huisken and Ilmanen [HI]. Using this flow they were able to
prove the Riemannian Penrose inequality of general relativity.
Here we present the results of our method for the numerical computation
of the IMCF in asymptotic flat manifolds. The method itself together with
some theoretical results can be found in [P], where one can also find some
remarks on implementation issues.
The motion of a surface can be best experienced in movies. Therefore
we have computed the following MPEG movies:
Examples in Euclidean space
Examples in other spaces
Resources for further information:
Deformation of a torus in Euclidean space
Suppose you start the IMCF with a thin torus in R^{3} with positive
mean curvature. The torus will become thicker and the mean curvature approaches
at some points zero. So the classical flow by inverse mean curvature, see
[G] and [U], cannot exist for infinite time.
In the weak formulation of [HI] the surface will "jump" to its minimizing
hull before this point is reached. The jump with infinite velocity occurs
if the area of the surface would be larger than the area of its minimizing
hull. After the jump the surface evolves in the classical fashion: in the
rescaled limit it approaches the unit sphere.
In our example the large radius of the torus equals 2.0 and the small
radius 0.5.
Movie (1591 kB)
Flow of two spheres in Euclidean space
For the weak formulation it doesn't matter if the starting surface is split
into two ore more components. So in this example we start with to unit
spheres. A classical solution for the IMCF exists up to the point where
the two expanding spheres would touch. As in the last example the weak
formulation jumps to the minimizing hull long before this point is reached.
Observe the catenoid that is formed as the connecting minimal surface
between the two spheres.
Movie (1483 kB)
Two tori in Euclidean space
The last two examples can be combined: we start with two tori in Euclidean
space and observe two topological changes of the initial surface. The first
jump transforms the tori into discs, whereas the second one inserts a catenoid
in between the
two discs.
Movie (504 kB)
Two spheres around two "black holes"
We use the following static metric, which is a modification of the classical
static Schwarzschild metric:
Both masses m_{1} and m_{2} are equal to one and x_{1}=(2,0,0),
x_{2}=(2,0,0). The starting surface is composed out of two spheres
of radius 0.55 around x_{1}, x_{2}.
Movie (small screen size, 213 kB)
Movie (large screen size, 751 kB)
Sphere around "black hole" near a static
star
Here we modify the conformal factor oft classical static Schwarzschild
metric with the conformal factor of the static metric of a star with
constant density rho:
If the density rho is small (1.0) the flow is only a little perturbed:
Movie (small screen size, 422 kB),
Movie
(large screen size, 1093 kB),
but if the density is large (5.0) we see a jump over 3/4 of the star:
Movie (small screen size, 270 kB),
Movie
(large screen size, 930 kB)
References

[G]: C. Gerhardt, "Flow of Nonconvex Hypersurfaces into Spheres", J. Diff.
Geom. 32, 1990, 299314

[HI]: G. Huisken and T. Ilmanen, "The
Inverse Mean Curvature Flow and the Riemannian Penrose Inequality",
preprint, 1998, to appear in J. Diff. Geom.

[HI1]: G. Huisken and T. Ilmanen, "A
Note on the Inverse Mean Curvature Flow", Saitama Workshop proceedings,
1998

[HI2]: G. Huisken and T. Ilmanen, "The
Riemannian Penrose Inequality", Int. Math. Res. Not. 20, 1997,
10451058

[P]: E. Pasch, "Numerische Verfahren zur Berechnung von Krümmungsflüssen",
Ph.D. thesis Universität Tübingen, 1998

[U]: J. Urbas, "On the expansion of starshaped hypersurfaces by symmetric
functions of their principal curvatures", Math. Z. 205, 1990, 355372
Links
pasch@na.unituebingen.de