In this expository talk, we discuss some inequalities holding
for certain solutions of Ricci flow. Ricci flow is a form of the heat
equation for Riemannian metrics. So techniques from the study of the
heat equation apply. Examples of basic inequalities include the
Li-Yau inequality for positive solutions of the heat equation, which
motivated the Harnack inequalities of Hamilton and Perelman for Ricci
flow. Fundamental inequalities of Perelman are for the entropy and for
the reduced volume. Moreover, there are many other inequalities which
hold for certain classes of solutions, such as those proved by
Hamilton, Perelman, and others.

The Ricci flow is an important tool in geometry, and a main
problem is to understand the stability of fixed points of the flow and the
convergence of solutions to those fixed points. There are many approaches
to this, but one involves maximal regularity theory and a theorem of
Simonett. I will describe this technique and its application to certain
extended Ricci flow systems. These systems arise from manifolds with
extra structure, such as fibration or warped product structures, or Lie
group structures.

Mandatory for 298 Grad Seminar students who have been a TA for less than 2 quarters at UCI

Mandatory for 298 Grad Seminar students who have been a TA for less than 2 quarters at UCI