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Constructing Kähler-Ricci solitons from Sasaki-Einstein manifolds. (English) Zbl 1222.53074

Kähler-Ricci solitons are self-similar solutions of the Kähler-Ricci flow. They are classified as expanding, steady, and shrinking for obvious reasons. The authors construct new Kähler-Ricci solitons from Sasaki-Einstein manifolds. Sasaki-Einstein manifolds are links of Ricci-flat Kähler cones and singularity models in Calabi-Yau manifolds. They first show that there is an expanding soliton flowing out of the Kähler cone over any Sasaki-Einstein manifold. The method they employ is the Calabi ansatz over Sasaki-Einstein manifolds. The same method is then applied to construct shrinking and expanding solitons on line bundles over Fano manifolds such that the associated \(U(1)\)-bundles admit Sasaki-Einstein metrics. Certain pairs of shrinking and expanding solitons can be pasted together to form an eternal solution of the Kähler-Ricci flow which lives on \((-\infty,\infty)\), with singularities along the zero section of the line bundle, but the shrinking solitons extend smoothly to the zero section when the Sasaki-Einstein structure is regular. The results generalize constructions of Cao and Feldman-Ilmanen-Knopf. We point out that the authors present examples which do not carry in general any continuous symmetry.

MSC:

53C55 Global differential geometry of Hermitian and Kählerian manifolds
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
55N91 Equivariant homology and cohomology in algebraic topology
53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)