Entropy and reduced distance for Ricci expanders
| dc.creator | Feldman, Michael | |
| dc.creator | Ilmanen, Tom | |
| dc.creator | Ni, Lei | |
| dc.date | 2004-05-03 | |
| dc.date.accessioned | 2026-07-07T05:07:53Z | |
| dc.date.available | 2026-07-07T05:07:53Z | |
| dc.description | Perelman has discovered two integral quantities, the shrinker entropy $\cW$ 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} $\ctW$ 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 $μ_+$ and $ν_+$. The {\it forward reduced volume} $θ_+$ is monotone in general and constant exactly on expanders. A natural conjecture asserts that $g(t)/t$ converges as $t\to\infty$ 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 $\Vol(g)/t^{n/2}$ (Hamilton) and $\barλ$ (Perelman), as well as our new quantities. In general, we show that if $\Vol(g)$ grows like $t^{n/2}$ (maximal volume growth) then $\ctW$, $θ_+$ and $\barλ$ remain bounded (in their appropriate ways) for all time. We attempt a sharp formulation of the conjecture. | |
| dc.identifier | https://arxiv.org/abs/math/0405036 | |
| dc.identifier | http://arxiv.org/abs/math/0405036 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/71044 | |
| dc.subject | Differential Geometry | |
| dc.subject | 58G11 | |
| dc.title | Entropy and reduced distance for Ricci expanders | |
| dc.type | text |