Gruber P. Convex and Discrete Geometry

200 Convex Bodies
vector of St at x(s, t) with speed equal to the curvature of St at x(s, t). In other
words, x(s, t) is the solution of the parabolic initial value problem
xt(s, t) = xss(s, t)
x(s, 0) = x(s),
where x(s) is a given arc-length parametrization of S0.
This problem, clearly, can be extended to dimensions d ≥ 2 with the flow driven
by the mean curvature, the Gauss curvature, a function of these, or by some other
function of the principal curvatures. For simplicity, we consider only the mean and
the Gauss curvature.
The following result was proved for d = 2 by Gage [349, 350] and Gage and
Hamilton [351] and for d ≥ 3 by Huisken [529]. Andrews [31] gave a generalization
providing a simpler proof of Huisken’s result. For the question of singularities which
might evolve, see Huisken and Sinestrari [530] and White [1024]. For a selection of
generalizations, related material and references, we refer to Andrews [33], the book
of Chou and Zhu [210] and the reports of White [1022, 1023].
Theorem 10.4. A given (sufficiently differentiable) closed convex surface S0 in E d
is deformed by the mean curvature driven flow in the interior normal direction into
a family {St : t ≥ 0} of closed convex surfaces shrinking to a point. If rescaled by
suitable homotheties, these convex surfaces tend to S d−1 .
An analogous result holds for the flow driven by the Gauss curvature. For d = 2
it is the same result as before. For general d it was proved by Chou [209] and Andrews
[32]. See also Andrews [33] for a multitude of references. For affine evolutions
compare Leichtweiss [643].
Convex Billiards and Caustics
A(convex) billiard table B in E d is a proper convex body (see Fig. 10.1). A billiard
ball in B is a point which moves with constant speed along a (straight) line in int B
until it hits bd B. If the point where it hits bd B is a regular boundary point of B,the
billiard ball is reflected in the usual way and moves again with the same speed along
a line in int B, etc. If the billiard ball hits bd B at a singular point, it stops there. The
curve described by a billiard ball is called a billiard trajectory.
Fig. 10.1. Billiard