ABSTRACT
Chapter 8
The Non-convex
Generalized Curve
Shortening Flow
In this chapter, we study the Cauchy problem for the generalized
curve shortening ow (GCSF),
@
@t
= jkj
kn ; (; 0) =
; (8.1)
where
is a C
-embedded closed curve. We shall investigate whether
the ow exists until it shrinks to a point and whether the Grayson
convexity theorem still holds. The main diÆculty is that the equa-
tion is singularndegenerate parabolic. The main result of this chapter
is the following almost convexity theorem.
