On Rotationally Symmetric Hamilton's Equation for Kähler-Einstein Metrics

Koiso, Norihito

doi:10.1016/b978-0-12-001018-9.50015-4

Cited by 138 publications

(136 citation statements)

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“…This forces ϕ to be a linear function in s. In the following we will show that this leads to gradient Kähler Ricci solitons generalising those constructed in [7,8,21,26]. As this analysis parallels that in [46], we shall be brief and only emphasise the necessary additional considerations.…”

Section: A Class Of Equidistant Hypersurface Familiesmentioning

confidence: 92%

“…From the study of Tian and Zhu [43,44], we note that Kähler-Ricci solitons on compact complex manifolds are unique up to holomorphic transformations. Several authors [1,7,8,10,21,23,26,33] have produced cohomogeneity one type Kähler-Ricci solitons where the hypersurfaces are circle bundles over a Fano Kähler-Einstein space. The hypersurfaces are equipped with metrics such that the bundle projection becomes a Riemannian submersion with totally geodesic fibres.…”

Section: Introductionmentioning

confidence: 99%

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On Ricci solitons of cohomogeneity one

2010

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We analyse some properties of the cohomogeneity one Ricci soliton equations, and use Ansätze of cohomogeneity one to produce new explicit examples of complete Kähler Ricci solitons of expanding, steady and shrinking types. These solitons are foliated by hypersurfaces which are circle bundles over a product of Fano Kähler-Einstein manifolds or over coadjoint orbits of a compactly connected semisimple Lie group.

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Section: A Class Of Equidistant Hypersurface Familiesmentioning

confidence: 92%

Section: Introductionmentioning

confidence: 99%

On Ricci solitons of cohomogeneity one

2010

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“…The study of Ricci solitons becomes important because of their relationship with singularities of the Ricci flow; they represent extremal cases for the Harnack estimates and may be limiting cases for the Ricci flow near singularities (see [Ca3,Ha3]). Just a few examples of Ricci solitons are known, see [Ca1,Ca2,Ca3,I4,Ko] for the Kähler case and see [Ha2,Br,I2] for the complete case. T. Ivey proved in [I1] the non-existence of compact 3-dimensional Ricci solitons and he gave in [I3] some results on local existence and on the dimension of the corresponding moduli space.…”

Section: Ricci Solitons and Quasi-einstein Metricsmentioning

confidence: 99%

Ricci soliton homogeneous nilmanifolds

Lauret¹

2001

Math Ann

192

211

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We study a notion weakening the Einstein condition on a left invariant Riemannian metric g on a nilpotent Lie group N . We consider those metrics satisfying Ric g = cI + D for some c ∈ R and some derivation D of the Lie algebra n of N, where Ric g denotes the Ricci operator of (N, g). This condition is equivalent to the metric g to be a Ricci soliton. We prove that a Ricci soliton left invariant metric on N is unique up to isometry and scaling. The following characterization is also given: (N, g) is a Ricci soliton if and only if (N, g) admits a metric standard solvable extension whose corresponding standard solvmanifold (S,g) is Einstein. This gives several families of new examples of Ricci solitons. By a variational approach, we furthermore show that the Ricci soliton homogeneous nilmanifolds (N, g) are precisely the critical points of a natural functional defined on a vector space which contains all the homogeneous nilmanifolds of a given dimension as a real algebraic set. (2000): 53C30, 53C21, 53C25, 22E25. Mathematics Subject Classification

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“…See for example: Cao [3]; Chave and Valent [4,5]; Feldman et al [7]; Hamilton [9,10]; Ivey [11,12]; Koiso [13] Pedersen et al [17]. We refer the reader to a survey article of Derdzinski [6], for more references on compact Ricci solitons i.e.…”

Section: Introductionmentioning

confidence: 99%

Conformal and Quasi-Einstein Metrics on Pseudo-Euclidean Space

Souza¹,

Tenenblat

2009

Results. Math.

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We consider constant symmetric tensors T on R n , n ≥ 3, and we study the problem of finding metrics ¯ g conformal to the pseudo-Euclidean metric g such that Ric ¯ g = T. We show that such tensors are determined by the diagonal elements and we obtain explicitly the metrics ¯ g. As a consequence of these results we get solutions globally defined on R n for the equation −ϕ∆gϕ + n|||gϕ|| 2 /2 + λϕ 2 = 0. Moreover, we show that for certain unbounded functions K defined on R n , there are metrics conformal to the pseudo-Euclidean metric with scalar curvature K.

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On Rotationally Symmetric Hamilton's Equation for Kähler-Einstein Metrics

Cited by 138 publications

References 4 publications

On Ricci solitons of cohomogeneity one

On Ricci solitons of cohomogeneity one

Ricci soliton homogeneous nilmanifolds

Conformal and Quasi-Einstein Metrics on Pseudo-Euclidean Space

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