Kähler immersions of Kähler–Ricci solitons into definite or indefinite complex space forms

@article{Loi2020KhlerIO,
  title={K{\"a}hler immersions of K{\"a}hler–Ricci solitons into definite or indefinite complex space forms},
  author={Andrea Loi and Roberto Mossa},
  journal={Proceedings of the American Mathematical Society},
  year={2020}
}
  • A. LoiR. Mossa
  • Published 2 March 2020
  • Mathematics
  • Proceedings of the American Mathematical Society
Let (g,X) be a Kähler–Ricci soliton on a complex manifold M . We prove that if the Kähler manifold (M, g) can be Kähler immersed into a definite or indefinite complex space form then g is Einstein. Notice that there is no topological assumptions on the manifold M and the Kähler immersion is not required to be injective. Our result extends the result obtained in [3] asserting that a KRS on a compact Kähler submanifold M ⊂ CP which is a complete intersection is KE. 
View PDF on arXiv
11 Citations
Background Citations
5
Methods Citations
1

11 Citations

Kähler–Ricci solitons induced by infinite-dimensional complex space forms

We exhibit families of non trivial (i.e. not Kähler-Einstein) radial Kähler-Ricci solitons (KRS), both complete and not complete, which can be Kähler immersed into infinite dimensional complex space…

On the non-existence of common submanifolds of Kähler manifolds and complex space forms

Two Kähler manifolds are called relatives if they admit a common Kähler submanifold with their induced metrics. In this paper, we provide a sufficient condition to determine whether a real analytic…

Extremal Kähler Metrics Induced by Finite or Infinite-Dimensional Complex Space Forms

In this paper, we address the problem of studying those complex manifolds M equipped with extremal metrics g induced by finite or infinite-dimensional complex space forms. We prove that when g is…

Umehara algebra and complex submanifolds of indefinite complex space forms

  • Xu ZhangD. Ji
  • Mathematics
    Annals of Global Analysis and Geometry
  • 2022
The Umehara algebra is studied with motivation on the problem of the non-existence of common complex submanifolds. In this paper, we prove some new results in Umehara algebra and obtain some…

Holomorphic isometries into homogeneous bounded domains

. 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…

Extremal Kaehler metrics induced by finite or infinite dimensional complex space forms

In this paper we address the problem of studying those complex manifolds $M$ equipped with extremal metrics $g$ induced by finite or infinite dimensional complex space forms. We prove that when $g$…

On canonical radial Kaehler metrics

We prove that a radial Kähler metric g is Kähler-Einstein if and only if one of the following conditions is satisfied: 1. g is extremal and it is associated to a Kähler-Ricci soliton; 2. two…

The Umehara algebra 2 . 1 The statement of the main theorem

The Umehara algebra is studied with motivation on the problem of the non-existence of common complex submanifolds. In this paper, we prove some new results in Umehara algebra and obtain some…

Sasakian immersions of Sasaki-Ricci solitons into Sasakian space forms

  • G. Placini
  • Mathematics
    Journal of Geometry and Physics
  • 2021

On holomorphic isometries into blow-ups of $\mathbb C^n$

. We study the Kähler-Einstein manifolds which admits a holomorphic isometry into either the generalized Burns-Simanca manifold (˜ C n ,g S ) or the Eguchi-Hanson manifold (˜ C 2 ,g EH ) . Moreover,…

References

SHOWING 1-10 OF 38 REFERENCES

Kähler–Ricci solitons on compact complex manifolds with C1(M) > 0

Abstract.In this paper, we discuss the relation between the existence of Kähler–Ricci solitons and a certain functional associated to some complex Monge–Ampère equation on compact complex manifolds…

Kähler-Ricci solitons on homogeneous toric bundles

Abstract We deal with homogeneous toric bundles M over generalized flag manifolds GC /P, where G is a compact semisimple Lie group and P a parabolic subgroup. Using symplectic data, we provide a…

Kahler-Ricci solitons on toric manifolds with positive first Chern class

Kähler–Einstein submanifolds of the infinite dimensional projective space

This paper consists of two main results. In the first one we describe all Kähler immersions of a bounded symmetric domain into the infinite dimensional complex projective space in terms of the…

Isometric embeddings of Kähler-Ricci solitons in the complex projective space

We prove that a compact complex manifold endowed with a nontrivial Kahler-Ricci soliton cannot be isometrically embedded in the FubiniStudy complex projective space as a complete intersection.

Kähler Immersions of Kähler Manifolds into Complex Space Forms

The study of K\"ahler immersions of a given real analytic K\"ahler manifold into a finite or infinite dimensional complex space form originates from the pioneering work of Eugenio Calabi [10]. With a…

Differential geometry of indefinite complex submanifolds in indefinite complex space forms

Classically, a Kaehler structure consists of a Riemann metric and a complex structure, which are related by well known compatibility conditions. The Riemann metric is then called a Kaehler metric. If…

A new holomorphic invariant and uniqueness of Kähler-Ricci solitons

Abstract. In this paper, a new holomorphic invariant is defined on a compact Kähler manifold with positive first Chern class and nontrivial holomorphic vector fields. This invariant generalizes the…

A note on kähler — Einstein metrics and Bochner's coordinates

In this paper we prove that if a compact Kähler-Einstein manifold(M, ω with integral Kahler form satisfies a compatibility condition between the domain of definition of the Bochner coordinates and of…

Projectively induced rotation invariant Kähler metrics

We classify Kähler–Einstein manifolds admitting a Kähler immersion into a finite dimensional complex projective space endowed with the Fubini–Study metric, whose codimension is less than or equal to…