On the long-time behavior of type-III Ricci flow solutions

By J. Lott. 2005

162 citations recorded by the source corpus.

We show that three-dimensional homogeneous Ricci flow solutions that admit finite-volume quotients have long-time limits given by expanding solitons. We show that the same is true for a large class of four-dimensional homogeneous solutions. We give an extension of Hamilton’s compactness theorem that does not assume a lower injectivity radius bound, in terms of Riemannian groupoids. Using this, we show that the long-time behavior of type-III Ricci flow solutions is governed by the dynamics of an $${\mathbb{R}}^+$$ -action on a compact space.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/07628ee91b470a01af304e4d504fdf63e4e1c757

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Ricci solitons: the equation point of view (2008, background): “REMARK 3.3. In the noncompact case, there exists non gradient Ricci solitons, see Baird and Danielo [1], Lott [ 21 ].”
Algebraic Ricci Solitons of three-dimensional Lorentzian Lie groups (2011, background): “In fact, any left-invariant Riemannian metric on the three-dimensional Heisenberg group is an expanding non-gradient Ricci soliton which is not an Einstein metric (see [1], [14], [17]).”
On Gradient Ricci Solitons with Symmetry (2007, result): “This is in sharp contrast to the more general Ricci solitons that exist on many Lie groups and other homogeneous spaces see [1, 14, 15].”
Generic Properties of Homogeneous Ricci Solitons (2007, methodology): “In this article the authors study 3-dimensional Ricci solitons which projects via a semi-conformal mapping to a surfaces and obtain a complete description of the soliton structures on Nil 3 and Sol 3 predicted by Lauret. In [ 11 ], Lott”
Ricci flow coupled with harmonic map flow (2009, background): “More advanced explanations of Perelman’s proof of the two conjectures are given in Cao and Zhu [3] Chow et al. [5, 6], Kleiner and Lott [19] and Morgan and Tian [25, 26].”
Canonical measures and Kähler-Ricci flow (2008, background): “It is discovered and intensively studied in [ Lo ].”
Dimensional reduction and the long-time behavior of Ricci flow (2007, methodology): “The notion of smooth pointed convergence of Riemannian groupoids is given in [38], which extends these collapsing considerations to the Ricci flow.”
Infinite time singularities of the Kähler-Ricci flow (2014, methodology): “Homogeneous type-III solutions were studied by Lott [15].”