Ricci flow and Einstein metrics in low dimensions (Q2771294)

This essay gives an expository account of Hamilton's work on Ricci surfaces [\textit{R. S. Hamilton}, Contemp. Math. 71, 237-261 (1988; Zbl 0663.53031)], 3-manifolds [J. Differ. Geom. 17, 255-306 (1982; Zbl 0504.53034)] and 4-manifolds [J. Differ. Geom. 24, 153-179 (1986; Zbl 0628.53042)] which are important in the theory of Ricci flow. NEWLINENEWLINENEWLINEIn Section 2 the basic facts about Ricci flow are presented: the equation for the time evolution, short-time existence and the equations for the geometric quantities associated to the metric, such as the Christoffel symbols, the Riemannian curvature tensor, Ricci tensor and scalar curvature. NEWLINENEWLINENEWLINESection 3 contains several maximum principles that are required for the study of the Ricci flow: for the scalar heat equation on a manifold and for systems where the solution is a section of a vector bundle. Both the weak and the strong maximum principle are considered. NEWLINENEWLINENEWLINEIn sections 4 through 6 the author devotes one section each to Hamilton's results on surfaces, 3-manifolds and 4-manifolds. The two-dimensional Ricci solitons which are fixed points of the normalized Ricci flow, the entropy and the Harnack estimated for the scalar curvature function under the normalized Ricci flow are also discussed. NEWLINENEWLINENEWLINEThe last section contains a list of some important recent work on Ricci flow not included in this paper.NEWLINENEWLINEFor the entire collection see [Zbl 0961.00021].