As on arXiv
Abstract
We prove two rigidity theorems on holomorphic isometries into homogeneous bounded domains. The first shows that a Kähler-Ricci soliton induced by the homogeneous metric of a homogeneous bounded domain is trivial, i.e. Kähler-Einstein. In the second one we prove that a homogeneous bounded domain and the flat (definite or indefinite) complex Euclidean space are not relatives, i.e. they do not share a common Kähler submanifold (of positive dimension). Our theorems extend the results proved in [A. Loi, R. Mossa, Proc. Amer. Math. Soc. 149 (2021), no. 11, 4931-4941] and [X. Cheng, Y. Hao, Ann. Global Anal. Geom. 60 (2021), no. 1, 167-180] respectively.
Comments: 11 pages. arXiv admin note: text overlap with arXiv:2003.00841
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Connected papers
The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2021, plus the canon.
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Rigidity properties of holomorphic isometries into homogeneous Kähler manifolds
Topics Kähler–Einstein & K-stability, Kähler–Ricci solitons, Einstein metrics · Same authors Andrea Loi, Roberto Mossa · Shared terms “homogeneous bounded”, “holomorphic isometries”, “bounded domain”
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Universal embeddings of flag manifolds and rigidity phenomena
Topics Kähler–Ricci solitons, Homogeneous spaces, Kähler geometry · Same authors Andrea Loi, Roberto Mossa · Shared terms “homogeneous bounded”, “bounded domain”, “holomorphic isometries”
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On the Bergman metric of Cartan-Hartogs domains
Topics Kähler–Einstein & K-stability, Kähler–Ricci solitons, Einstein metrics · Same authors Andrea Loi, Roberto Mossa · Shared terms “bounded domain”, “Cheng”
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Immersions of Sasaki-Ricci solitons into homogeneous Sasakian manifolds
Topics Kähler–Ricci solitons, Homogeneous spaces, Kähler geometry · Same author Roberto Mossa · Shared terms “homogeneous bounded”, “holomorphic isometries”, “bounded domain”
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Ricci iterations of well-behaved Kähler metrics
Topics Kähler–Einstein & K-stability, Kähler–Ricci solitons, Einstein metrics · Same author Andrea Loi · Shared terms “theorems extend”
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On canonical radial Kaehler metrics
Topics Kähler–Einstein & K-stability, Kähler–Ricci solitons, Einstein metrics · Same author Andrea Loi
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The Kähler-Ricci soliton on bounded pseudoconvex domains
Topics Kähler–Einstein & K-stability, Kähler–Ricci solitons, Einstein metrics · Shared terms “Cheng”
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Extremal Kähler–Ricci solitons on Fano manifolds are Kähler–Einstein
Topics Kähler–Einstein & K-stability, Kähler–Ricci solitons, Einstein metrics
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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
- 11A.13Homogeneous Flows
A computable laboratory: the flow on Thurston’s geometries.
- 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.