Perelman defined his W-functional and proved the entropy monotonicity formulae for Hamilton's Ricci flow.
The critical points of W-functional are shrinking gradient Ricci solitons (SGRS). It is well known that gradient Ricci solitons are generalizations of Einstein manifolds and basic models for smooth metric measure spaces. In this talk, I will discuss some recent progress and problems in four dimensional cases. In particular, one of the challenging problems is to classify all gradient Ricci solitons with constant scalar curvature. Recently in a joint work with X. Cheng, we prove that a 4-dimensional shrinking gradient Ricci soliton has constant scalar curvature if and only if it is either Einstein, or a finite quotient of Gaussian shrinking soliton R^4, S^2×R^2 or S^3×R.
周德堂，巴西Fluminense联邦大学教授，已发表论文80余篇，包括JDG, Math. Ann., Crelle’s Journal, IMRN, CVPDE, JFA, Trans. AMS等国际知名期刊。