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 [2], Wei and Wylie [13], Limoncu [4, 5], Rimoldi [7] and Zhang [14].
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 [6], quasi-Einstein manifolds [1] and Sasaki--Ricci solitons [3].
References
[1] J. Case, Y.-J. Shu and G. Wei, Rigidity of quasi--Einstein metrics, Diff. Geom. Appl. 29 (2011)
[2] M. Fernández López and E. García Río, A remark on compact Ricci solitons, Math. Ann. 340 (2008)
[3] A. Futaki, H. Ono and G. Wang, Transverse Kähler geometry of Sasaki manifolds and toric Sasaki-Einstein manifolds, J. Differential Geom. 83 (2009)
[4] M. Limoncu, Modifications of the Ricci tensor and applications, Arch. Math. (Basel) 95 (2010)
[5] -, The Bakry-Emery Ricci tensor and its applications to some compactness theorems, Math. Z. 271 (2012)
[6] S. Pigola, M. Rigoli, M. Rimoldi and A. G. Setti, Ricci almost solitons, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), Vol. X (2011)
[7] M. Rimoldi, A remark on Einstein warped products, Pacific J. Math. 252 (2011)
[8] H. Tadano, Some Myers Type Theorems and Hitchin--Thorpe Inequalities for Shrinking Ricci Solitons, to appear in Rend. Semin. Mat. Univ. Politec. Torino (2016)
[9] -, Remark on a diameter bound for complete Riemannian manifolds with positive Bakry-Emery Ricci curvature, Diff. Geom. Appl. 44 (2016)
[10] -, An upper diameter bound for compact Ricci solitons with applications to the Hitchin-Thorpe inequality, arXiv:1504.05577 (2015)
[11] -, Diameter bounds, gap theorems and Hitchin-Thorpe inequalities for compact quasi-Einstein manifolds, in preparation (2016)
[12] -, Gap theorems for compact gradient Sasaki-Ricci solitons, Internat. J Math. 26 (2015), 15400091 [13] G. Wei and W. Wylie, Comparison geometry for the Bakry-Emery Ricci tensor, J. Differential Geom. 83 (2009) [14] S. Zhang, A theorem of Ambrose for Bakry-Emery Ricci tensor, Ann. Global Anal. Geom. 45 (2014)
© Homare Tadano.