MONDAY, AUGUST 19, 2024

This talk is a survey to the Li-Yau inequality and its applications in the Ricci flow. In 1986, Li-Yau studied the heat equation on manifolds and found a fundamental important inequality for the positive solutions. It implies a sharp estimate to the heat kernels and gives the monotonicity of the associated weighted volume. 

Later, I discussed with Hamilton on its generalization to tensors on the Ricci flow. The first thing we agreed was to rewrite the inequality to be in quadratic form. Note that the Li-Yau inequality becomes an equality on the heat kernels on Euclidean spaces. Then try to test it on all Ricci solitons which is the extreme case of the inequality. This was finally worked out by Hamilton after a couple of years.

In 2002-2003, Perelman extended the Li-Yau inequality to the conjugate heat equation. The monotonicity of the corresponding volume gives the noncollapsing estimates of the Ricci flow. Perelman also used Hamilton's Li-Yau inequality to understand the structure of singularities and control the surgeries of the Ricci flow. These are key ingredients in the proof of the Poincaré and geometrization conjectures.