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 the Ricci flow. We prove a gap theorem for ancient solutions, which could be regarded as an entropy counterpart of Yokota's work. In addition, we prove that under some assumptions on one time slice of a complete ancient solution with nonnegative curvature operator, finite asymptotic entropy implies kappa-noncollapsing on all scales. This result is used by the author [21] to prove Perelman's assertion that on an ancient solution to the Ricci flow with bounded nonnegative curvature operator, bounded entropy is equivalent to noncollapsing on all scales; see section 11 in [17].