The Ricci Flow: Techniques and Applications

Chow, Bennett; Chu, Sun-Chin; Glickenstein, David; Guenther, Christine; Isenberg, James; Ivey, Tom; Knopf, Dan; Lü, Peng; Luo, Feng; Ni, Lei

doi:10.1090/surv/135

Cited by 182 publications

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“…For a (Kähler-)Ricci soliton there are some quantities that are constant, see e.g. [9]. One of them is, in our notation, (10) s + |∇f | 2 + 2f = const .…”

Section: Statements and Proofsmentioning

confidence: 99%

“…These metrics give rise to special solutions of the Kähler-Ricci flow (see e.g. [9]), namely they evolve under biholomorphisms. It is known that on a compact manifold, if c ≥ 0 then g is Einstein (see e.g.…”

Section: Introductionmentioning

confidence: 99%

“…It is known that on a compact manifold, if c ≥ 0 then g is Einstein (see e.g. again [9]), so in the compact Kähler case one only considers the so-called shrinking Kähler-Ricci solitons (c < 0) whose equation, after a scaling, can be written as…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On Calabi extremal Kähler-Ricci solitons

Calamai

Petrecca

2015

Proc. Amer. Math. Soc.

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“…For a (Kähler-)Ricci soliton there are some quantities that are constant, see e.g. [9]. One of them is, in our notation, (10) s + |∇f | 2 + 2f = const .…”

Section: Statements and Proofsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On Calabi extremal Kähler-Ricci solitons

Calamai

Petrecca

2015

Proc. Amer. Math. Soc.

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“…We now discuss a straightforward application of the above symmetry of the vacuum Einstein equations and produce new self-similar solutions of the Ricci flow with two commuting hypersurfaceorthogonal Killing vector fields. For this we only review the basic definitions and readers are referred to standard references for more details [13,14].…”

Section: Ricci Flow and Ricci Solitonsmentioning

confidence: 99%

Static axisymmetric Einstein equations in vacuum: Symmetry, new solutions, and Ricci solitons

Akbar

MacCallum

2015

Phys. Rev. D

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An explicit one-parameter Lie point symmetry of the four-dimensional vacuum Einstein equations with two commuting hypersurface-orthogonal Killing vector fields is presented. The parameter takes values over all of the real line and the action of the group can be effected algebraically on any solution of the system. This enables one to construct particular one-parameter extended families of axisymmetric static solutions and cylindrical gravitational wave solutions from old ones, in a simpler way than most solution-generation techniques, including the prescription given by Ernst for this system [17]. As examples, we obtain the families that generalize the Schwarzschild solution and the C-metric. These in effect superpose a Levi-Civita cylindrical solution on the seeds. Exploiting a correspondence between static solutions of Einstein's equations and Ricci solitons (self-similar solutions of the Ricci flow), this also enables us to construct new steady Ricci solitons.

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“…We recall that Bryant constructed a steady Ricci soliton as the warped product (0, +∞) × f S m , m > 1, with a radial warping function f . Bryant did not himself publish this result, but it can be checked in [13]. Since a warped product is a complete manifold for all warping function if and only if both base and fiber are complete manifolds (see Bishop and O'Neill [5] or O'Neill [22]), the difficulty in the construction of Bryant was to show the completeness of his example.…”

Section: Introductionmentioning

confidence: 99%

Gradient Ricci almost soliton warped product

Feitosa

Filho

Gomes

et al. 2019

Journal of Geometry and Physics

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We present the necessary and sufficient conditions for constructing gradient Ricci almost solitons that are realized as warped products. This will be done by means of Bishop-O'Neill's formulas and a particular study of Riemannian manifolds satisfying a Ricci-Hessian type equation. We prove existence results and give an example of particular solutions of the PDEs that arise from our construction. We also prove a rigidity result for a gradient Ricci soliton Riemannian product in the class of gradient Ricci almost soliton warped products under some natural geometric assumptions on the warping function.2010 Mathematics Subject Classification. Primary 53C15, 53C25; Secondary 53C21.

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The Ricci Flow: Techniques and Applications

Cited by 182 publications

References 0 publications

On Calabi extremal Kähler-Ricci solitons

On Calabi extremal Kähler-Ricci solitons

Static axisymmetric Einstein equations in vacuum: Symmetry, new solutions, and Ricci solitons

Gradient Ricci almost soliton warped product

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