In this talk, we shall establish some Cheeger-Gromov-Taylor type compactness theorems via modified Ricci curvature. Our compactness theorems generalize the classical compactness theorem by Cheeger, Gromov and Taylor  and improve the Myers type theorem via modified Ricci curvature by Limoncu .
 J. Cheeger, M. Gromov and M. Taylor, Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds, J. Differential Geom. 17 (1982), 15-53.
 M. Limoncu, Modifications of the Ricci tensor and applications, Arch. Math. (Basel) 95 (2010), 191-199.
© Homare Tadano.