Entropy and reduced distance for Ricci expanders

Perelman has discovered two integral quantities, the shrinker entropy $\text{cW}$ and the (backward) reduced volume, that are monotone under the Ricci flow $\pa g_{ij}/\pa t=-2R_{ij}$ and constant on shrinking solitons. Tweaking some signs, we find similar formulae corresponding to the expanding case. The {\it expanding entropy} $\text{ctW}$ 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 ${\mu }_{+}$ and ${\nu }_{+}$. The {\it forward reduced volume} ${\theta }_{+}$ is monotone in general and constant exactly on expanders. A natural conjecture asserts that $g(t)\mathrm{/}t$ converges as $t\to \mathrm{\infty }$ 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 $\text{Vol}(g)\mathrm{/}{t}^{n\mathrm{/}2}$ (Hamilton) and $\bar{\lambda }$ (Perelman), as well as our new quantities. In general, we show that if $\text{Vol}(g)$ grows like $t^{n\mathrm{/}2}$ (maximal volume growth) then $\text{ctW}$, ${\theta }_{+}$ and $\bar{\lambda }$ remain bounded (in their appropriate ways) for all time. We attempt a sharp formulation of the conjecture.