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.

The squared-mean-curvature integral was introduced two centuries
ago by Sophie Germain to model the bending energy and vibration patterns of
thin elastic plates. By the 1920s the Hamburg geometry school realized
this energy is invariant under the Möbius group of conformal
transformations, and thus that minimal surfaces in R^3 or S^3 are
equilibria. In 1965 Willmore observed that the round sphere minimizes the
bending energy among all closed surfaces, and he conjectured that a certain
torus of revolution - the stereographic projection of the Clifford minimal
torus in S^3 - minimizer for surfaces of genus 1. This conjecture was
proven this spring by Fernando Coda and Andre Neves; they use the
Almgren-Pitts minimax construction, the Hersch-Li-Yau notion of conformal
area, and Urbano's characterization of the Clifford torus by its Morse
index. We will discuss what is known or conjectured for other topological
types of bending-energy minimizing or equilibrium surfaces, and how
bending-energy gradient flow might be applied to problems in
low-dimensional topology.