ABSTRACT
Invariant Solutions for the
Curve Shortening Flow
We discuss invariant solutions { travelling waves, spirals and self-
similar solutions { for the generalized curve shortening ow (GCSF)
@
@t
=
k
kn ; > 0 : (2.1)
They are not merely examples. In fact, they can be used as com-
parison functions to yield a priori estimates. Even more signicant
is their role in the classication of the singularities of (2.1). For the
curve shortening ow, the travelling waves, which are called grim
reapers, describe the asymptotic shape of type II singularity and the
contracting self-similar solutions (they are circles when embedded)
characterize type I singularities. We shall discuss this in Chapter
5. Travelling waves, spirals, and expanding self-similar solutions ap-
pear to be very stable. They can be used to describe the long time
behaviour of complete, unbounded solutions of (2.1).
