ABSTRACT
Chapter 1
Basic Results
In this chapter, we rst establish the existence of a maximal solution
and some basic qualitative behaviour such as the separation princi-
ple and niteness of nodes for the general ow (1.2). These proper-
ties are direct consequences of the parabolic nature of the ow. For
the reader's convenience, we collect fundamental results on parabolic
equations in Section 2. In particular, the \Sturm oscillation theo-
rem," which is not found in standard texts on this subject, will play
an important role in the removal of singularities of the ow. In Sec-
tion 3, we derive evolution equations for various geometric quantities
of the ow. They will become important when we study the long time
behaviour of the ow.
