Entropy, noncollapsing, and a gap theorem for ancient solutions to the Ricci flow

In this paper we discuss the asymptotic entropy for ancient solutions to Ricci flow. We prove a gap theorem solutions, which could be regarded as an counterpart of Yokota's work. addition, that under some assumptions on one time slice complete solution with nonnegative curvature operator, finite implies kappa-noncollapsing all scales. This provides evidence Perelman's more general assertion bounded is equivalent kappa-noncollapsing.