In this talk, after reviewing some recent compactness theorems for Ricci solitons [4, 5, 7], we shall give some new compactness theorems via m-modified Ricci and m-Bakry-Émery Ricci curvatures with negative m. Our results may be considered as natural generalizations of the classical compactness theorems due to Ambrose [1], Cheeger-Gromov-Taylor [2], and Galloway [3] and may be compared with the Myers type theorem due to Wylie [8].
We shall also give a new condition for compact four-dimensional Ricci solitons to satisfy the Hitchin-Thorpe inequality. This condition sharpens the previous condition given by the speaker [6]. If time permits, we shall discuss about a compactness theorem for complete Sasaki manifolds.
References
[1] W. Ambrose, Duke Math. J. 24 (1957).
[2] J. Cheeger, M. Gromov, and M. Taylor, J. Differential Geom. 17 (1982).
[3] G. J. Galloway, J. Differential Geom. 14 (1979).
[4] M. Limoncu, Arch Math. (Basel), 95 (2010).
[5] Y. Soylu, to appear in Diff. Geom. Appl. (2017).
[6] H. Tadano, J. Math. Phys. 58 (2017).
[7] G. Wei and W. Wylie, J. Differential Geom. 83 (2009).
[8] W. Wylie, Geom. Dedicata 178 (2015).
© Homare Tadano.