Paper · Sep 2, 2019

Quasi-Einstein hypersurfaces of complex space forms

Xiaomin Chen

math.DGarXiv:1909.00753
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Abstract

Based on a well-known fact that there are no Einstein hypersurfaces in a non-flat complex space form, in this article we study the quasi-Einstein condition, which is a generalization of an Einstein metric, on the real hyersurface of a non-flat complex space form. For the real hypersurface with quasi-Einstein metric of a complex Euclidean space, we also give a classification. Since a gradient Ricci soliton is a special quasi-Einstein metric, our results improve some conclusions of.

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Submitted
Sep 2, 2019
Subjects
math.DG

Comments: 13 pages. arXiv admin note: text overlap with arXiv:1710.06794

Ricci solitonsEinstein metricsAlmost & η-Ricci solitonsSphere theorems

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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 2016, plus the canon.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper Real hypersurfaces of complex quadric in terms of star-Ricci tensor (Xiaomin Chen, 2017)5Chen 2017 Real hypersurfaces with ^(*)-Ricci solitons of non-flat complex space forms (Xiaomin Chen, 2017)1Chen 2017 Volume growth estimates for Ricci solitons and quasi-Einstein manifolds (Xu Cheng, Ernani Ribeiro and Detang Zhou, 2020)7Cheng 2020 A Study On Some Geometric Properties and Physical Applications of A Mixed Quasi-Einstein Spacetime (Kaushik Chattopadhyay, Arindam Bhattacharyya and Dipankar Debnath, 2021)8Chattopadhya 2021 Gradient pseudo-Ricci solitons of real hypersurfaces (Mayuko Kon, 2021)4Kon 2021 The extended quasi-Einstein manifolds with generalised Ricci solitons (Zhiming Huang, Weijun Lu and Fuhong Su, 2022)6Huang 2022 Conformal Ricci solitons on generalized (κ, μ)-space forms (Mehraj Ahmad Lone and Towseef Ali Wani, 2023)2Lone 2023 Quasi-Einstein Metrics and a curvature identity associated with the Ricci flow (Atreyee Bhattacharya and Sayoojya Prakash, 2025)3Bhattacharya 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Real hypersurfaces with -Ricci solitons of non-flat complex space forms

    Xiaomin Chen

    Topics Sphere theorems, Ricci solitons · Same author Xiaomin Chen · Shared terms “complex space”, “flat complex”, “space form”

  2. Conformal Ricci solitons on generalized ()-space forms

    Mehraj Ahmad Lone and Towseef Ali Wani

    Topics Almost & η-Ricci solitons, Sphere theorems, Einstein metrics · Shared terms “space form”, “gradient Ricci”

  3. Quasi-Einstein Metrics and a curvature identity associated with the Ricci flow

    Atreyee Bhattacharya and Sayoojya Prakash

    Topics Almost & η-Ricci solitons, Einstein metrics, Ricci solitons · Shared terms “quasi Einstein”, “gradient Ricci”

  4. Gradient pseudo-Ricci solitons of real hypersurfaces

    Mayuko Kon

    Topics Sphere theorems, Ricci solitons · Shared terms “real hypersurfaces”, “complex space”, “space form”

  5. Real hypersurfaces of complex quadric in terms of star-Ricci tensor

    Xiaomin Chen

    Topics Einstein metrics, Ricci solitons · Same author Xiaomin Chen · Shared terms “real hypersurfaces”

  6. The extended quasi-Einstein manifolds with generalised Ricci solitons

    Zhiming Huang, Weijun Lu and Fuhong Su

    Topics Almost & η-Ricci solitons, Einstein metrics, Ricci solitons · Shared terms “quasi Einstein”

  7. Volume growth estimates for Ricci solitons and quasi-Einstein manifolds

    Xu Cheng, Ernani Ribeiro and Detang Zhou

    Topics Almost & η-Ricci solitons, Einstein metrics, Ricci solitons · Shared terms “quasi Einstein”, “gradient Ricci”

  8. A Study On Some Geometric Properties and Physical Applications of A Mixed Quasi-Einstein Spacetime

    Kaushik Chattopadhyay, Arindam Bhattacharyya and Dipankar Debnath

    Topics Almost & η-Ricci solitons, Einstein metrics, Ricci solitons · Shared terms “quasi Einstein”

Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) Einstein metrics Metrics whose Ricci curvature is a constant multiple of the metric. They are the fixed points of normalised Ricci flow, and the most natural metrics… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) Almost & η-Ricci solitons Variations on the Ricci soliton equation: let the constant λ become a function (almost solitons), add a term along a contact form (η-Ricci… Start with: Ricci almost solitons (Pigola et al., 2010) Sphere theorems Curvature conditions that force a manifold to be a sphere. Ricci flow settled the 1/4-pinched differentiable sphere theorem, which had been open for… Start with: Minimal Two-Spheres and the Topology of Manifolds with Positive Curvature on Totally Isotropic Two-Planes (Micallef et al., 1988)

In the guidebook

  1. 11A.11
    Hamilton’s 1982 TheoremBook 11A · The Ricci Flow: Existence and Maximum Principles

    Positive Ricci curvature in dimension 3 flows to a round metric.

  2. 11B.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

    Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.

  3. 12B.2
    The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery

    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.