It is shown that the diameter of a compact shrinking Ricci soliton has a universal lower bound. This is proved by extending universal estimates for the first non-zero eigenvalue of Laplacian on compact Riemannian manifolds with lower Ricci curvature bound to a twisted Laplacian on compact shrinking Ricci solitons.
"Lower diameter bounds for compact shrinking Ricci solitons." Asian J. Math. 17 (1) 17 - 32, March 2013.