Ricci flow with surgery on three-manifolds

By G. Perelman. 2003

1,439 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Cellular and Molecular Neuroscience, Radiology, Nuclear Medicine and Imaging

View Paper: https://www.semanticscholar.org/paper/5a82c581b26bcabff7a9655a23126571d8c30af5

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In Applied Mathematics

In Cellular and Molecular Neuroscience

In Radiology, Nuclear Medicine and Imaging

Citations in the corpus

On the volume functional of compact manifolds with boundary with constant scalar curvature (2008, background): “This conjecture remained widely open until recently its 3-dimensional case followed as a corollary of Perelman’s work on geometrization [14, 15].”
Gradient estimates for solutions of the heat equation under Ricci flow (2009, neutral): “DG/0211159 [Perelman 2003] G. Perelman, “Ricci flow with surgery on three manifolds”, preprint, 2003. arXiv math.”
Global attractor and repeller of Morse-Smale diffeomorphisms (2010, methodology): “The original Poincar´ e conjecture was proved in 2002 by Perelman [41, 42 ].”
Dark Energy from structure: a status report (2007, methodology): “[154], [ 155 ]) on the Ricci‐Hamilton flow rel ated to the recent proof of Poincar´ e’s conjecture, and implied progress on Th urston’s geometrization program [5] to cut a Riemannian manifold into ‘nice pieces’ o f eight elementary geometries.”
Estimates for the extinction time for the Ricci flow on certain 3-manifolds and a question of Perelman (2003, background): “The next theorem gives an upper bound for the derivative of W (g(t)) under the Ricci flow which forces the solution g(t) to become extinct in finite time (see paragraph 4.4 of [ 11 ] for the precise definition of extinction time when surgery occurs).”
Mean Curvature Flow of Mean Convex Hypersurfaces (2013, background): “A similar issue (ruling out the possibility of collapsing with bounded curvature) arises in the Ricci flow, where it was handled by Perelman’s W-functional 1 In fact, White proves most of his results in a setting that also allows general ambient manifolds and nonsmooth initial conditions: (1) still ”
Noncompact Shrinking 4-Solitons with Nonnegative Curvature (2007, methodology): “Under assumptions of nnnegative curvature andκ-noncollapseness Perelman classified three dimensional gradient shrinking solitons [14] for the purpose of studying such singularities.”
Distinguishing geometries using finite quotients (2014, methodology): “This is one case of the geometrization theorem, proved by Perelman [37; 38; 39] (see also Kleiner and Lott [28] and Morgan and Tian [33; 34]).”