Topic

# Ricci flow

About: Ricci flow is a research topic. Over the lifetime, 3198 publications have been published within this topic receiving 73400 citations. The topic is also known as: Ricci-Hamilton flow.

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TL;DR: In this article, a monotonic expression for Ricci flow, valid in all dimensions and without curvature assumptions, is presented, interpreted as an entropy for a certain canonical ensemble.

Abstract: We present a monotonic expression for the Ricci flow, valid in all dimensions and without curvature assumptions. It is interpreted as an entropy for a certain canonical ensemble. Several geometric applications are given. In particular, (1) Ricci flow, considered on the space of riemannian metrics modulo diffeomorphism and scaling, has no nontrivial periodic orbits (that is, other than fixed points); (2) In a region, where singularity is forming in finite time, the injectivity radius is controlled by the curvature; (3) Ricci flow can not quickly turn an almost euclidean region into a very curved one, no matter what happens far away. We also verify several assertions related to Richard Hamilton's program for the proof of Thurston geometrization conjecture for closed three-manifolds, and give a sketch of an eclectic proof of this conjecture, making use of earlier results on collapsing with local lower curvature bound.

3,091 citations

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3,014 citations

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TL;DR: In this paper, the Ricci form of some Kahler metric is shown to be closed and its cohomology class must represent the first Chern class of M. This conjecture of Calabi can be reduced to a problem in non-linear partial differential equation.

Abstract: Therefore a necessary condition for a (1,l) form ( G I a ' r r ) I,,, Rlr dz' A d? to be the Ricci form of some Kahler metric is that it must be closed and its cohomology class must represent the first Chern class of M. More than twenty years ago, E. Calabi [3] conjectured that the above necessary condition is in fact sufficient. This conjecture of Calabi can be reduced to a problem in non-linear partial differential equation.

2,903 citations

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TL;DR: In this article, the Ricci flow with surgeries was constructed, and a lower bound on the volume of maximal horns and the smoothness of solutions was established. But this lower bound was later shown to be unjustified and irrelevant for the other conclusions.

Abstract: This is a technical paper, which is a continuation of math.DG/0211159. Here we construct Ricci flow with surgeries and verify most of the assertions, made in section 13 of that e-print: the exceptions are (1) the statement that manifolds that can collapse with local lower bound on sectional curvature are graph manifolds - this is deferred to a separate paper, since the proof has nothing to do with the Ricci flow, and (2) the claim on the lower bound for the volume of maximal horns and the smoothness of solutions from some time on, which turned out to be unjustified and, on the other hand, irrelevant for the other conclusions.

1,200 citations

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TL;DR: In this article, the structure of spaces Y which are pointed Gromov Hausdor limits of sequences f M i pi g of complete connected Riemannian manifolds whose Ricci curvatures have a de nite lower bound was studied.

Abstract: In this paper and in we study the structure of spaces Y which are pointed Gromov Hausdor limits of sequences f M i pi g of complete connected Riemannian manifolds whose Ricci curvatures have a de nite lower bound say RicMn i n In Sections and sometimes in we also assume a lower volume bound Vol B pi v In this case the sequence is said to be non collapsing If limi Vol B pi then the sequence is said to collapse It turns out that a convergent sequence is noncollapsing if and only if the limit has positive n dimensional Hausdor measure In par ticular any convergent sequence is either collapsing or noncollapsing Moreover if the sequence is collapsing it turns out that the Hausdor dimension of the limit is actually n see Sections and Our theorems on the in nitesimal structure of limit spaces have equivalent statements in terms of or implications for the structure on a small but de nite scale of manifolds with RicMn n Al though both contexts are signi cant for the most part it is the limit spaces which are emphasized here Typically the relation between corre sponding statements for manifolds and limit spaces follows directly from the continuity of the geometric quantities in question under Gromov Hausdor limits together with Gromov s compactness theorem Theorems see also Remark are examples of

1,031 citations