Blog

A central problem of differential geometry is the geometrization problem on manifolds. In particular, it is to determine which manifolds admit certain geometric structures. One of methods is to understand and classify the singularity models of the corresponding nonlinear geometric evolution equation, and to connect it to the existence problem of geometric structures on manifolds. In 1982, R. Hamilton introduced the Ricci flow and then by studying the singularity models of Ricci flow, G. Perelman completely solved Thurston geometrization conjecture and Poincare conjecture for a closed 3-manifold in 2002 and 2003. On the other hand, Mori minimal model program in birational geometry can be viewed as the complex analogue of Thurston’s geometrization conjecture. In 1985, H.-D. Cao introduced the Kähler-Ricci flow and then recaptured the Calabi-Yau Conjecture. Moreover, there is a conjecture picture by Song-Tian that the Kähler-Ricci flow should carry out the analytic minimal model program with scaling on projective varieties. Recently, Song-Weinkove established the above conjecture on a projective algebraic surface. Furthermore, via the Sasaki-Ricci flow, Chang-Lin-Wu proved the Sasaki analogue of minimal model program on closed quasi-regular Sasakian 5-manifolds. In this lecture, I will address the above issues in some detail.