On the asymptotic reduced volume of the Ricci flow

Abstract

Abstract. In this paper, we consider two different monotone quantities defined for the Ricci flow and show that their asymptotic limits coincide for any ancient solutions. One of the quantities we consider here is Perelman's reduced volume, while the other is the local quantity discovered by Ecker, Knopf, Ni and Topping. This establishes a relation between these two monotone quantities.

On Ecker's local integral quantity at infinity for ancient mean curvature flows

“…Observing above facts, we come up with a question about the relation between these integrals at infinity. Inspired by the work of Yokota [11] for ancient solution of Ricci flow, we expect that the above two monotone quantities coincide with each other at infinity. In fact, this is true if we assume…”

“…The author would like to thank Yohei Sakurai for helpful discussions during this work. The author is grateful to Takumi Yokota for giving him a rough idea of the proof in [11] for Ricci flow.…”

“…In this section, we give the proof of Theorem 1.3. We follow the same line as in [5] and [11] (see also [3]). The proof is divided into several steps.…”

Liouville theorem for heat equation along ancient super Ricci flow via reduced geometry

“…We first review basics of reduced geometry. The references are [8], [31], [32], [38], [39], [40]. We mainly refer to the works of Yokota [39], [40], which is compatible with our setting.…”

Ricci Flow Approach to The Cosmological Constant Problem

“…The convergence of V∞ and u 0 are crucial facts which relate to the renormalizability and asymptotic safety of the Q-NLSM, leading to a convergent value of CC. Actually, V∞ (g) is the Gaussian density Θ(g) [62][63][64] of a UV manifolds, its Ln can be given by a limit of the W-functional introduced also by Perelman [19,65,66],…”