The formations of singularities in the Ricci Flow

Hamilton, Richard S.

doi:10.4310/sdg.1993.v2.n1.a2

Cited by 709 publications

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“…Extensive research has been done in the case of the smooth flow (cf. [22], [6], [14], [21], [25], [12], [13], [30], or [15] for complete updated references). In order to prove the uniqueness of extremal Kähler metrics in full generality, in [9], the first two named authors were led to the study of Kähler-Ricci flows in the weak sense.…”

Section: Introductionmentioning

confidence: 99%

On the weak Kähler-Ricci flow

Chen

Тиан

Zhang

2011

Trans. Amer. Math. Soc.

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Section: Introductionmentioning

confidence: 99%

On the weak Kähler-Ricci flow

Chen

Тиан

Zhang

2011

Trans. Amer. Math. Soc.

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“…It is known by results of R.Hamilton ([6] [8]) and W.Shi that the Ricci flow exists and preserve the above two conditions. R.Hamilton asks if the Ricci flow exists globally.…”

Section: Introductionmentioning

confidence: 99%

“…The positive curvature condition is for the injectivity radius bound used for the compactness theorem [8] [11].…”

Section: Introductionmentioning

confidence: 99%

Injectivity radius bound of Ricci flow with positive Ricci curvature and applications

Ма

Zhu

2013

Front. Math. China

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Abstract. In this paper, we study the injectivity radius bound for 3-d Ricci flow. We also give an example showing that the positivity of Ricci curvature can not be weakened into non-negativity. As applications we show the long time existence of the Ricci flow with positive Ricci curvature. We also partially settle a question in page 302 (line -13) of the book of Chow-Lu-Ni (2006).Mathematics Subject Classification 2000: 53Cxx,35Jxx

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“…In fact, as has become standard in discussions of Ricci flow, following Simon [43], we use in place of Ricci flow a generalization of the DeTurck gauging (DeTurck [9]) called the dual Ricci-Harmonic Map flow (see Hamilton [17], §6 or Simon [43], p. 3 for a precise discussion), which is equivalent to the Ricci flow up to a diffeomorphism. In what follows, we refer simply to "flow".…”

Section: Theorem 56 (Schoen [40]) Let S Be a Stable Minimal Surfacementioning

confidence: 99%

Positivity of the universal pairing in 3 dimensions

Calegari

Freedman

Walker

2009

J. Amer. Math. Soc.

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Associated to a closed, oriented surface S S is the complex vector space with basis the set of all compact, oriented 3 3 -manifolds which it bounds. Gluing along S S defines a Hermitian pairing on this space with values in the complex vector space with basis all closed, oriented 3 3 -manifolds. The main result in this paper is that this pairing is positive, i.e. that the result of pairing a nonzero vector with itself is nonzero. This has bearing on the question of what kinds of topological information can be extracted in principle from unitary ( 2 + 1 ) (2+1) -dimensional TQFTs. The proof involves the construction of a suitable complexity function c c on all closed 3 3 -manifolds, satisfying a gluing axiom which we call the topological Cauchy-Schwarz inequality, namely that c ( A B ) ≤ max ( c ( A A ) , c ( B B ) ) c(AB) \le \max (c(AA),c(BB)) for all A , B A,B which bound S S , with equality if and only if A = B A=B . The complexity function c c involves input from many aspects of 3 3 -manifold topology, and in the process of establishing its key properties we obtain a number of results of independent interest. For example, we show that when two finite-volume hyperbolic 3 3 -manifolds are glued along an incompressible acylindrical surface, the resulting hyperbolic 3 3 -manifold has minimal volume only when the gluing can be done along a totally geodesic surface; this generalizes a similar theorem for closed hyperbolic 3 3 -manifolds due to Agol-Storm-Thurston.

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The formations of singularities in the Ricci Flow

Cited by 709 publications

References 0 publications

On the weak Kähler-Ricci flow

On the weak Kähler-Ricci flow

Injectivity radius bound of Ricci flow with positive Ricci curvature and applications

Positivity of the universal pairing in 3 dimensions

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