Ricci solitons were introduced by Hamilton in 1982 and are natural generalizations of Einstein manifolds. They correspond to self-similar solutions to the Ricci flow and often arise as singularity models. The importance of Ricci solitons was demonstrated in papers by Perelman, where Ricci solitons played crucial roles in his affirmative resolution of the Poincaré conjecture. Besides their geometric importance, Ricci solitons are also of great interest in theoretical physics and have been studied actively in relation to string theory.
In this talk, we shall establish some compactness criteria and diameter estimates for complete shrinking Ricci solitons. Our compactness theorems generalize previous ones obtained by Fernández-López and García-Río , Wei and Wylie , Limoncu [4, 5], Rimoldi  and Zhang .
As applications of these compactness theorems, we shall give some upper diameter bounds for compact Ricci solitons. Moreover, by using such diameter bounds, we shall provide some new sufficient conditions for four-dimensional compact Ricci solitons to satisfy the Hitchin-Thorpe inequality. If time permits, we shall also consider some natural generalizations of Ricci solitons such as Ricci almost solitons , quasi-Einstein manifolds  and Sasaki--Ricci solitons .
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 M. Fernández López and E. García Río, A remark on compact Ricci solitons, Math. Ann. 340 (2008)
 A. Futaki, H. Ono and G. Wang, Transverse Kähler geometry of Sasaki manifolds and toric Sasaki-Einstein manifolds, J. Differential Geom. 83 (2009)
 M. Limoncu, Modifications of the Ricci tensor and applications, Arch. Math. (Basel) 95 (2010)
 -, The Bakry-Emery Ricci tensor and its applications to some compactness theorems, Math. Z. 271 (2012)
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 -, An upper diameter bound for compact Ricci solitons with applications to the Hitchin-Thorpe inequality, arXiv:1504.05577 (2015)
 -, Diameter bounds, gap theorems and Hitchin-Thorpe inequalities for compact quasi-Einstein manifolds, in preparation (2016)
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© Homare Tadano.