Perelman's entropy on ancient Ricci flows

Abstract

In [ZY2], the second author proved Perelman's assertion, namely, for an ancient solution to the Ricci flow with bounded and nonnegative curvature operator, bounded entropy is equivalent to noncollapsing on all scales. In this paper, we continue this discussion. It turns out that the curvature operator nonnegativity is not a necessary condition, and we need only to assume a consequence of Hamilton's trace Harnack. Furthermore, we show that this condition holds for steady Ricci solitons with nonnegative Ricci curvature.

Ancient Ricci flows with asymptotic solitons

“…Proposition 4.1 (Corollary 5.11 in [Bam20a]; c.f. Corollary 4.5 in [MZ21]). Let (M n , g(t)) t∈I be a complete Ricci flow with bounded curvature within each time interval compact in I.…”

“…The estimates therein within the smooth category can be regarded as some good improvements of Perelman's entropy techniques, whereas the weak version of the Ricci flow (called the metric flow ) provides an intriguing object for the future study. While the latter is fascinating and promising, it is yet beyond the scope of the present article, and we shall focus on the former, just as the last two authors have done in [MZ21].…”

“…The last two authors [MZ21] have observed that, on an ancient solution with bounded curvature within each compact time interval, the limit of the Nash entropy lim τ →∞ N x,t (τ ) and the limit of Perelman's entropy lim τ →∞ W x,t (τ ) are independent of the base point (x, t). What is new in the corollary below is that the limit of the reduced volume lim τ →∞ V x,t (τ ) is also independent of the base point (x, t).…”

“…What is new in the corollary below is that the limit of the reduced volume lim τ →∞ V x,t (τ ) is also independent of the base point (x, t). This fact follows not from [MZ21], but from an observation that in the statement of Assumption B, the base point (p 0 , 0) is not so important as the sequence of positive scales {τ i } ∞ i=1 . Corollary 1.2.…”

“…Obviously, we may also use Theorem 1.4 to obtain a finer version of the main results in [Zh20] and [MZ21]. We include the following corollary, and the details of the proof are merely combinations of Theorem 1.4 with [Zh20] and [MZ21].…”