## Abstract

Perelman has discovered two integral quantities, the shrinker entropy W and the (backward) reduced volume, that are monotone under the Ricci flow ∂_{gij}/∂t = − 2R_{ij} and constant on shrinking solitons. Tweaking some signs, we find similar formulae corresponding to the expanding case. The expanding entropy W_{+} is monotone on any compact Ricci flow and constant precisely on expanders; as in Perelman, it follows from a differential inequality for a Harnack-like quantity for the conjugate heat equation, and leads to functionals μ_{+} and v_{+}. The forward reduced volume θ_{+} is monotone in general and constant exactly on expanders.

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 expanders; these include vol(g)/t^{n/2} (Hamilton) and -λ (Perelman), as well as our new quantities. In general, we show that, if vol(g) grows like t^{n/2}(maximal volume growth) then W^{+}, θ^{+} and -λ remain bounded (in their appropriate ways) for all time. We attempt a sharp formulation of the conjecture.

## Math Subject Classifications

58G11## Key Words and Phrases

Entropy Ricci flow Ricci expanders asymptotic behavior reduced volume monotonicity Li-Yau-Hamilton inequality## Preview

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