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Entropy and reduced distance for Ricci expanders. (English) Zbl 1071.53040

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

MSC:

53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
35K05 Heat equation
28D20 Entropy and other invariants
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