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 , Cheeger-Gromov-Taylor , and Galloway  and may be compared with the Myers type theorem due to Wylie .
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 . If time permits, we shall discuss about a compactness theorem for complete Sasaki manifolds.
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© Homare Tadano.