Entropy and reduced distance for Ricci expanders
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Perelman has discovered two integral quantities, the shrinker entropy W and the (backward) reduced volume, that are monotone under the Ricci flow ∂gij/∂t = − 2Rij 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)/tn/2 (Hamilton) and -λ (Perelman), as well as our new quantities. In general, we show that, if vol(g) grows like tn/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 Classifications58G11
Key Words and PhrasesEntropy Ricci flow Ricci expanders asymptotic behavior reduced volume monotonicity Li-Yau-Hamilton inequality
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