 
Summary: Proceedings of the Conference on Elliptic and Parabolic Equations,
held at Gregynog, August 1989.
Shrinking Doughnuts.
SIGURD B. ANGENENT.
Introduction.
Let Mn
be a smooth compact oriented manifold, and let X : M ×
[0, T ) Rn+1
be a smooth family of immersions of M in n+1 dimensional
Euclidean space. The orientation of M allows one to define a unique smooth
unit normal vector field X : M × [0, T ) Rn+1
. Given this choice of X,
we can define the principal curvatures, 1, . . . , n, of the immersion X(·, t)
and the mean curvature HX = (1 + . . . + n)/n in the usual way. By
definition, the family of immersions X(·, t) "moves by its mean curvature"
if the normal velocity satisfies
(1) Xt(p, t), X (p, t) = nHX(p, t)
for all (p, t) M × [0, T ). Here x, y = x0y0 + · · · + xnyn denotes the
Euclidean inner product on Rn+1
.
