Ricci Flow and the Poincaré Conjecture

By Siddhartha Gadgil, Harish Seshadri. 2007

208 citations recorded by the source corpus.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/6c396fc471073061460e840500a3f8d9c4ae8fb6

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Citations in the corpus

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].”
Topological methods in moduli theory (2014, methodology): “8 The flow method made then its way further through the work of Hamilton [206], Perelman and others [320–322], leading to the solution of the three dimensional Poincaré conjecture (see for example [299] for an exposition).”
3-Manifold Groups (2012, methodology): “We refer to [MTi07] for full details and to [CZ06a, CZ06b, KlL08, BBBMP10] for further information on the proof.”
Geometrisation of 3-Manifolds (2010, methodology): “As in [51], we use the following definition.”
A Compactness Theorem for Complete Ricci Shrinkers (2010, unknown): “The entropy was introduced by Perelman in his famous paper [30] to solve the long standing problem of ruling out collapsing with bounded curvature (see [4, 9, 25, 28] for detailed expositions of Perelman’s work).”
The maximal number of exceptional Dehn surgeries (2008, unknown): “Thus, now that Perelman’s proof of this conjecture [33–35] is accepted as correct [24, 29, 30], a compact orientable 3-manifold is hyperbolike if and only if it is hyperbolic.”
Conjectures on Bridgeland stability for Fukaya categories of Calabi-Yau manifolds, special Lagrangians, and Lagrangian mean curvature flow (2014, unknown): “Be warned that I expect the difficulty of proving Conjectures 3.2 and 3.34 increases sharply with dimension, and even in dimension 3 is probably comparable in difficulty to the three-dimensional Poincaré Conjecture, as proved by Perelman and others (see Morgan and Tian [54]).”
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]).”
The minimal volume orientable hyperbolic 2-cusped 3-manifolds (2008, unknown): “The arguments of [2, 3] depend strongly on Perelman’s proof of the geometrization conjecture (see [19, 18, 12, 14, 6, 15]).”
A review of numerical methods for nonlinear partial differential equations (2012, unknown): “The Ricci ﬂow , introduced by Hamilton [98] and used by Perelman [164, 153] to solve the Poincar´e conjecture, takes the form (2.10) Here the unknown w ≡ w αβ ( t, x ) is a 3 × 3 time-dependent array of a Riemannian metric on a manifold M , and Ric = Ric αβ is the Ricci curvature tensor associated…”
Notes on Perelman's papers (2006, methodology): “After the May 2006 version of these notes was posted on the ArXiv, expositions of Perelman’s work appeared in [15] and [45].”
Noncompact shrinking four solitons with nonnegative curvature (2010, unknown): “Under various curvature assumptions similar estimates may be found in [14], [9], Chapter 6.”
Short-time behaviour of a modified Laplacian coflow of G2-structures (2012, unknown): “55) From this, it is possible to obtain an expression for Ric (the details can be found in [5, 22], for example) in terms of the background quantities and difference h̄ Ric = Ric− 1 2 ∇̄(2)h+ 1 2 LV̄ ḡ +O ( |h|(2) )”