ABSTRACT
A Class of Non-convex
Anisotropic Flows
In this chapter, we continue the study of ows for non-convex curves.
Let and be two smooth, 2-period functions of the tangent angle
satisfying
() > 0 (6.1)
and
( + ) = () ; ( + ) = () : (6.2)
We consider the Cauchy problem for
@
@t
= (k + )n ; (6.3)
where the initial curve
is a smooth, embedded closed curve. This
ow may be regarded as the linear case for the general ow (1.2),
where F is uniformly parabolic and symmetric. Remember that the
condition (6.2) means that F is symmetric. Without this condition,
embeddedness may not be preserved under the ow. When
is
convex, we have shown in x3.2 that the ow also preserves convexity
and it shrinks to a point where ! is nite. In this chapter we shall
show that the Grayson convexity theorem holds for (6.3).
