Entropy and reduced distance for Ricci expanders

Abstract

ABSTRACT. Perelman has discovered two integral quantities, the shrinker entropy A natural conjecture asserts that g(t)/t converges as t → ∞ to a negative Einstein manifold in some weak sense (in particular ignoring collapsing parts). If the limit is known a-priori to be smooth and compact, this statement follows easily from any monotone quantity that is constant on expandersSmall, large and distant parts of a Ricci flow are known to be modeled by various kinds of Ricci solitons: Steady, shrinking, and expanding. Perelman has discovered monotone quantities corresponding to the first two of these. The functional F is monotone on the Ricci flow and constant precisely on steadies [25, Section 1]. The shrinking, or localizing, entropy W is monotone in general and constant precisely on shrinking solitons [25, Section 3]. Closely related is the notion of the (backward) reduced volume [25, Section 7]. The latter two show that developing singularities and ancient histories are modeled on shrinkers, when an appropriate blowup, or blowdown is taken [25,27].In this note, by tweaking some signs, we observe similar monotonicity formulae for the expanding case. In Section 1, analogous to [25, Section 1], we define an expanding, or delocalizing entropy W + , which is monotone in general and constant precisely on expanders. It follows from Math Subject Classifications. Primary: 58G11.

Optimal transport and Perelman’s reduced volume

“…In this section we give the analogs of Sections 6 and 7 for the L + -functional that was considered in [6]. This is for possible future reference.…”

“…In fact, there are three relevant costs for Ricci flow : one corresponding to Perelman's L-functional (which we will call L − ), one corresponding to the Feldman-Ilmanen-Ni L + -functional [6] and a third one which we call L 0 . In the case of the L − -cost, the main result of the paper is the following.…”

Eigenvalues and energy functionals with monotonicity formulae under Ricci flow

“…Remark 63 In [6], M. Feldman, T. Ilmanen, and L. Ni constructed a scale invariant W-entropy which is an analogue of Perelman's W-entropy but has vanishing first variation over expanders. There is also a very good unified treatment about entropy formulae over steady, expanding, and shrinking Ricci breathers in [4].…”

The second variation of the Ricci expander entropy

“…By [Feldman et al 2005, Theorem 1.7], we know that µ + (g, σ ) is attained by some function f . Moreover, if λ(g) < 0, then ν + (g) can be attained by some positive number σ .…”

“…To find the corresponding variational structure for the expanding case, M. Feldman, T. Ilmanen and L. Ni [Feldman et al 2005] introduced the functional ᐃ + . Let (M n , g) be a compact Riemannian manifold, f a smooth function on M, and σ > 0.…”