Paper · Jan 21, 2025

Quaternionic Kähler manifolds fibered by solvsolitons

Vicente Cortés, Alejandro Gil-García, Markus Röser

math.DGarXiv:2501.12225
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Abstract

This paper is concerned with the geometry of principal orbits in quaternionic Kähler manifolds of cohomogeneity one. We focus on the complete cohomogeneity one examples obtained from the non-compact quaternionic Kähler symmetric spaces associated with the simple Lie groups of type A by the one-loop deformation. We prove that for zero deformation parameter the principal orbits form a fibration by solvsolitons (nilsolitons if ). The underlying solvable group is non-unimodular if and is the Heisenberg group if . We show that under the deformation, the hypersurfaces remain solvmanifolds but cease to be Ricci solitons.

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Submitted
Jan 21, 2025
Subjects
math.DG

Comments: 24 pages. Comments are welcome!

Ricci solitonsKähler geometryHomogeneous spacesBoundaries & non-compact

Its neighbourhood

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.

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 Inhomogeneous deformations of Einstein solvmanifolds (Adam Thompson, 2023)2Thompson 2023 A Hilbert–Mumford criterion for nilsolitons (Yoshinori Hashimoto, 2023)8Hashimoto 2023 Classification of superpotentials for cohomogeneity one Ricci solitons (Qiu Shi Wang, 2024)6Wang 2024 Ricci solitons as submanifolds of complex hyperbolic spaces (Ángel Cidre-Díaz and Víctor Sanmartín-López, 2024)3Cidre-Díaz 2024 Non-Shrinking Ricci Solitons of cohomogeneity one from the quaternionic Hopf fibration (Hanci Chi, 2024)5Chi 2024 3-symmetric spaces, Ricci solitons, and homogeneous structures (Thomas Murphy and Paul-Andi Nagy, 2025)1Murphy 2025 Mean curvature flow for principal orbits of Hermann actions on rank two symmetric spaces (Naoyuki Koike and Sakura Nakaoka, 2025)7Koike 2025 Generalized Ricci Solitons on Heisenberg and Oscillator Algebras (Vicente Cortés, Shubham Dwivedi and Lorenz Jetter, 2026)4Cortés 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. 3-symmetric spaces, Ricci solitons, and homogeneous structures

    Thomas Murphy and Paul-Andi Nagy

    Topics Homogeneous spaces, Boundaries & non-compact, Kähler geometry · Shared terms “symmetric space”, “simple Lie”, “Lie groups”

  2. Inhomogeneous deformations of Einstein solvmanifolds

    Adam Thompson

    Topics Homogeneous spaces, Ricci solitons · Shared terms “solvmanifolds”, “unimodular”, “cohomogeneity”

  3. Ricci solitons as submanifolds of complex hyperbolic spaces

    Ángel Cidre-Díaz and Víctor Sanmartín-López

    Topics Homogeneous spaces, Boundaries & non-compact, Ricci solitons · Shared terms “symmetric space”, “solvable”, “Lie”

  4. Generalized Ricci Solitons on Heisenberg and Oscillator Algebras

    Vicente Cortés, Shubham Dwivedi and Lorenz Jetter

    Topics Homogeneous spaces, Ricci solitons · Same author Vicente Cortés · Shared terms “Heisenberg”, “Lie groups”

  5. Non-Shrinking Ricci Solitons of cohomogeneity one from the quaternionic Hopf fibration

    Hanci Chi

    Topics Homogeneous spaces, Ricci solitons · Shared terms “quaternionic”, “cohomogeneity”, “fibration”

  6. Classification of superpotentials for cohomogeneity one Ricci solitons

    Qiu Shi Wang

    Topics Homogeneous spaces, Kähler geometry, Ricci solitons · Shared terms “cohomogeneity”

  7. Mean curvature flow for principal orbits of Hermann actions on rank two symmetric spaces

    Naoyuki Koike and Sakura NakaokaWider flows

    Topics Homogeneous spaces · Shared terms “principal orbits”, “symmetric space”, “cohomogeneity”

  8. A Hilbert–Mumford criterion for nilsolitons

    Yoshinori Hashimoto

    Topics Homogeneous spaces, Ricci solitons · Shared terms “nilsolitons”, “Lie groups”

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) Kähler geometry Complex manifolds with compatible Riemannian metrics. Here Ricci curvature is the complex Hessian of a log-determinant, so Ricci flow becomes a… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) Homogeneous spaces On homogeneous spaces Ricci flow reduces to an ordinary differential equation on a finite-dimensional space of invariant metrics. Start with: Ricci flow of locally homogeneous geometries on closed manifolds (Isenberg et al., 1992) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989)

In the guidebook

  1. 11A.13
    Homogeneous FlowsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A computable laboratory: the flow on Thurston’s geometries.

  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.