The Ricci Flow: An Introduction

By Bennett Chow, Dan Knopf. 2004

794 citations recorded by the source corpus.

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

View Paper: https://www.semanticscholar.org/paper/d902db00e816c101668e9b7b9293dee5caea26f2

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Citations in the corpus

Notes on Perelman's papers (2006, background): “Good sources for background material on Ricci flow are [22, 23, 33, 66].”
Certain Results on K-Contact and (k, μ)-Contact Manifolds (2008, methodology): “For details we refer to Chow and Knopf [ 6 ] and Derdzinski [7].”
On the classification of gradient Ricci solitons (2007, background): “page 186 of [ 10 ] for the derivation of the equivalence of these two formulas.”
Examples of Algebraic Ricci Solitons in the Pseudo-Riemannian Case (2011, background): “According to [9], we check that a Ricci soliton is a Ricci flow solution.”
Recent developments on the Hamilton’s Ricci Flow (2007, background): “The intuition of forming a Type II singularity on compact manifolds was described by Hamilton [47] (see also [28] and [99]) and the existence of a Type II singularity on compact manifolds was also proposed as an open question in the introduction of the book of Chow-Lu-Ni [29].”
Rigidity of shrinking Ricci solitons (2011, result): “We refer to [2,5–7] and the references therein for background on Ricci solitons.”
Backwards uniqueness of the Ricci flow (2009, neutral): “As in [CK], we define c(t) + λ(t − T ) + 1, and take φt : M → M to be the family of diffeomorphisms defined by ∂ ∂t φt(x) = 2X̃(x, t) + 2 c(t) X(φt(x)), φT = Id.”
Linear and dynamical stability of Ricci-flat metrics (2004, methodology): “where if A and B are two tensors, we denote by A ∗ B any quantity obtained from A ⊗ B by one of the operations explained in [ 6 ].”
A gradient estimate for all positive solutions of the conjugate heat equation under Ricci flow (2006, neutral): “(II) +(III) +(IV). We estimate each term on the right-hand side by the following: (I) = ψτ η(τ)q+ψητ q= ψr rτ η(τ)q+ψητ q. From the lower bound assumption on the Ricci curvature Ric≥ −K, we have (see [CK] e.g.) ∂r ∂τ = − ∂r ∂t (3.6) ≥ −Kr. By construction of ψ, we have ψr ≤ 0. Therefore, ψr ∂r ∂τ ≤ ”
The Space of Spaces: Curvature Bounds and Gradient Flows on the Space of Metric Measure Spaces (2012, neutral): “[CK04].”
On a classification of the gradient shrinking solitons (2007, background): “[ CK ], Theorem 7.1), we have that |rmRijkl| are uniformly bounded on M. Hence all the integrals involved are finite which then implies, via cut-o function argument, that the integrations by parts are completely legal, in view of the fast decay of e f ensured by Lemma 2.3 and the lower bound of S pr”
On the global well-posedness of energy-critical Schr\"odinger equations in curved spaces (2010, methodology): “, [Cao and Zhu 2006; Chow and Knopf 2004; Chow et al. 2007, 2008; 2010; Hamilton 1982; Morgan and Tian 2007; Shi 1989]) is given by @ @t Rijk` DRijk`C ijk`; (3-1) where”
Special Fast Diffusion with Slow Asymptotics: Entropy Method and Flow on a Riemannian Manifold (2008, background): “Note that for d = 2 we are dealing with an Einstein metric, R = k g (actually, it is Hamilton’s cigar soliton to the Ricci flow, [19, 11]), but for d ≥ 3 it is not.”