An equation linking 𝒲-entropy with reduced volume

Abstract

W -entropy and reduced volume for the Ricci flow were introduced by Perelman, which had proved their importance in the study of the Ricci flow. L. Ni studied the analogous concepts for the linear heat equation on the static manifolds, and established an equation which links the large time behavior of these two. Due to the surprising similarity between those concepts in the Ricci flow and the linear heat equation, a natural question whether such equation holds for the Ricci flow ancient solution was asked by L. Ni. In this paper, we gave an alternative proof to L. Ni's original equation based on a new method. And following the same philosophy of this method, we answer L. Ni's question positively for Type I κ-solutions of the Ricci flow.

Perelman's entropy on ancient Ricci flows

“…The proof of Theorem 1.9 together with some of Bamler's [Bam20a] results on the Nash entropy also implies the following interesting corollary, which is an extension of a result of Xu [Xu17] Corollary 1.11 (Entropy uniqueness of the asymptotic shrinker). Let (M, g(t)) t∈(−∞,0] be a κ-noncollapsed ancient solution with bounded curvature on each time-slice.…”

“…The main focus of Proposition 2.4 is the equality (2.5). Xu [Xu17] first proved it for Type I noncollapsed ancient solutions, and the second author [ZY2] proved it in the bounded and nonnegative curvature operator case.…”

On the noncollapsedness of positively curved Type I ancient Ricci flows

“…One may then follow the proofs in [6] or [22] to conclude this Proposition. Obviously, the noncollapsing condition in these proofs can be replaced by (3.4).…”

“…Proof. The proof of this proposition can be modified from, for instance, [6] or [22]. First of all, the Type I condition implies that there exists a positive number C, such that…”

Heat kernel estimates and asymptotic of W-entropy on stochastically complete Riemannian manifolds

“…He used the sharp pointwise bounds for the heat kernel proved by Li, Tam and Wang [11], which is closely related to the large time behavior of heat kernel by Li in [12] (also see [13] Main results. We first state logarithmic Sobolev inequality [15] on M with the constant depending on the optimal Euclidean constant (i.e., K(n, 2) 2 ∼ 2/πen ) as…”