Paper · Oct 31, 2024
Explicit complete Ricci-flat metrics and Kähler-Ricci solitons on direct sum bundles
As on arXiv
Abstract
Let be a Kähler-Einstein Fano manifold, and be a suitable root of the canonical bundle. We give a construction of complete Calabi-Yau metrics and gradient shrinking, steady, and expanding Kähler-Ricci solitons on the total space , of certain vector bundles , composed of direct sums of powers of . We employ the theory of hamiltonian 2-forms [2, 3] as an Ansatz, thus generalizing recent work of the author and Apostolov on [5], as well as that of Cao, Koiso, Feldman-Ilmanen-Knopf, Futaki-Wang, and Chi Li [10, 26, 23, 24, 30] when has Calabi symmetry. As a result, we obtain new examples of asymptotically conical Kähler shrinkers, Calabi-Yau metrics with ALF-like volume growth, and steady solitons with volume growth .
Comments: Minor improvements to the introduction and appendix. In particular, the curvature estimates in the appendix have been extended to apply in all cases of Theorem A
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Background reading
The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.
In the guidebook
- 11B.1Ricci Solitons
Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.
- 12B.2The Asymptotic Soliton and Three-Dimensional Shrinkers
Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.