ABSTRACT
In a truly remarkable paper [P1], Perelman discovered several quantities which are monotone under Ricci flow. These are the analytical breakthroughs that led him to the proof of Poincare´ and geometrization conjectures. One of the monotone quantities is the F entropy. Its monotonicity implies the monotonicity of the first eigenvalue of the operator −4∆ + R. This can also be regarded as a family of Poincare´ inequalities which are uniform in time. The other quantity is called the W entropy, which is a family of log Sobolev inequalities in disguise. Its monotonicity implies the crucial no local collapsing result under Ricci flow.
