Ricci flow on 4-manifolds and Seiberg-Witten equations (Q2900277)

The purpose of this article is to summarize the recent joint works of the authors and Y. Zhang on the long term solutions of the Ricci flow, based on the first author's talk in the ICCM.NEWLINENEWLINEThe content is as follows. First, the authors recall Perelman's entropy functionals and the limits of Ricci flows. Some gemetrical and topological constraints (e.g. the Hitchin-Thorpe and the Miyaoka-Yau inequalities) in dimension four for the underlying manifolds with long term solutions under various curvature assumptions are presented.NEWLINENEWLINEFor the main results, see the authors et al. [Math. Ann. 340, No. 3, 647--674 (2008; Zbl 1131.53034), Commun. Math. Phys. 283, No. 1, 1--24 (2008; Zbl 1155.53032); J. Geom. Anal. 20, No. 3, 592--608 (2010; Zbl 1198.53069)].NEWLINENEWLINEThe present paper contains many updated references and comments, it will be very useful for young researchers interested in this current topic.NEWLINENEWLINEFor the entire collection see [Zbl 1235.00045].