Entropy and reduced distance for Ricci expanders |
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Authors: | Michael Feldman Tom Ilmanen Lei Ni |
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Institution: | (1) Department of Mathematics, University of Wisconsin, 53706 Madison, WI;(2) Departement Mathematik, ETH Zentrum, 8092 Zürich, Switzerland;(3) Department of Mathematics, University of California, San Diego, 92093 La Jolla, CA |
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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 = − 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. |
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Keywords: | Math Subject Classifications" target="_blank">Math Subject Classifications 58G11 |
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