Paper · Jul 9, 2024

Ricci solitons as submanifolds of complex hyperbolic spaces

Ángel Cidre-Díaz, Víctor Sanmartín-López

math.DGarXiv:2407.06999
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Abstract

Any homogeneous expanding Ricci soliton is known to be isometric to a Lie subgroup of the solvable part of the Iwasawa decomposition associated with a symmetric space of non-compact type, with the metric induced as a submanifold. In this paper, we classify and analyze the geometry of such Lie subgroups with Ricci soliton induced metric when the symmetric spaces are complex hyperbolic spaces.

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Submitted
Jul 9, 2024
Subjects
math.DG
Ricci solitonsHomogeneous 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 Codimension one Ricci soliton subgroups of nilpotent Iwasawa groups (Victor Sanmartin-Lopez, 2022)1Sanmartin-Lo 2022 Long-time behavior of awesome homogeneous Ricci flows (Roberto Araujo, 2023)7Araujo 2023 3-symmetric spaces, Ricci solitons, and homogeneous structures (Thomas Murphy and Paul-Andi Nagy, 2025)2Murphy 2025 Quaternionic Kähler manifolds fibered by solvsolitons (Vicente Cortés, Alejandro Gil-García and Markus Röser, 2025)3Cortés 2025 The mean curvature flow of subgroups on Lie groups of dimension three (Romina M. Arroyo, Gabriela P. Ovando and Mariel Sáez, 2025)5Arroyo 2025 On steady and expanding Ricci solitons with asymptotic symmetries (Michael B. Law, 2025)4Law 2025 Einstein metrics and Killing spinors on pseudo-Riemannian solvmanifolds (Diego Conti, Federico A. Rossi and Romeo Segnan Dalmasso, 2025)6Conti 2025 Cohomogeneity One Expanding Ricci Solitons and the Expander Degree (Abishek Rajan, 2025)8Rajan 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Codimension one Ricci soliton subgroups of nilpotent Iwasawa groups

    Victor Sanmartin-Lopez

    Topics Homogeneous spaces, Boundaries & non-compact, Ricci solitons · Same author Víctor Sanmartín-López · Shared terms “Lie subgroup”, “Iwasawa”, “induced metric”

  2. 3-symmetric spaces, Ricci solitons, and homogeneous structures

    Thomas Murphy and Paul-Andi Nagy

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

  3. Quaternionic Kähler manifolds fibered by solvsolitons

    Vicente Cortés, Alejandro Gil-García and Markus Röser

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

  4. On steady and expanding Ricci solitons with asymptotic symmetries

    Michael B. Law

    Topics Homogeneous spaces, Boundaries & non-compact, Ricci solitons · Shared terms “expanding Ricci”

  5. The mean curvature flow of subgroups on Lie groups of dimension three

    Romina M. Arroyo, Gabriela P. Ovando and Mariel SáezWider flows

    Topics Homogeneous spaces · Shared terms “Lie subgroup”

  6. Einstein metrics and Killing spinors on pseudo-Riemannian solvmanifolds

    Diego Conti, Federico A. Rossi and Romeo Segnan Dalmasso

    Topics Homogeneous spaces, Ricci solitons · Shared terms “Iwasawa”, “hyperbolic space”, “Lie”

  7. Long-time behavior of awesome homogeneous Ricci flows

    Roberto Araujo

    Topics Homogeneous spaces, Boundaries & non-compact

  8. Cohomogeneity One Expanding Ricci Solitons and the Expander Degree

    Abishek Rajan

    Topics Homogeneous spaces, Ricci solitons · Shared terms “expanding Ricci”

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) 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.