Paper · Dec 9, 2022

Codimension one Ricci soliton subgroups of nilpotent Iwasawa groups

Victor Sanmartin-Lopez

math.DGarXiv:2212.04743
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As on arXiv

Abstract

Any expanding homogeneous Ricci soliton (in particular any homogeneous Einstein manifold of negative scalar curvature) can be obtained, up to isometry, from a Lie subgroup of a nilpotent Iwasawa group whose induced metric is a Ricci soliton. By nilpotent Iwasawa group we mean the nilpotent Lie group of the Iwasawa decomposition associated with a symmetric space of non-compact type. Motivated by this fact, in this paper we classify codimension one Lie subgroups of any nilpotent Iwasawa group whose induced metric is a Ricci soliton.

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Submitted
Dec 9, 2022
Subjects
math.DG
Ricci solitonsEinstein metricsScalar curvatureHomogeneous spacesBoundaries & non-compact

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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 2021, plus the canon.

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Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Ricci solitons as submanifolds of complex hyperbolic spaces

    Ángel Cidre-Díaz and Víctor Sanmartín-López

    Topics Homogeneous spaces, Boundaries & non-compact, Ricci solitons · Same author Victor Sanmartin-Lopez · Shared terms “Lie subgroup”, “Iwasawa”, “induced metric”

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

    Diego Conti, Federico A. Rossi and Romeo Segnan Dalmasso

    Topics Einstein metrics, Homogeneous spaces, Ricci solitons · Shared terms “Iwasawa”, “nilpotent”, “Lie”

  3. Infinitesimal maximal symmetry and Ricci soliton solvmanifolds

    Carolyn Gordon and Michael Jablonski

    Topics Einstein metrics, Homogeneous spaces, Ricci solitons · Shared terms “expanding homogeneous”, “Lie groups”, “isometry”

  4. 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”, “nilpotent”, “compact type”

  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”, “Lie groups”, “isometry”

  6. Equivariant solutions to the optimal partition problem for the prescribed Q-curvature equation

    Juan Carlos Fernández, Oscar Palmas and Jonatán Torres Orozco

    Topics Einstein metrics, Homogeneous spaces, Scalar curvature

  7. Solvable Lie algebras obtained by quivers and Einstein metrics

    Fumika Mizoguchi

    Topics Einstein metrics, Homogeneous spaces, Ricci solitons · Shared terms “nilpotent”, “Lie”

  8. On the Isometry Group of Immortal Homogeneous Ricci Flows

    Roberto Araujo

    Topics Homogeneous spaces, Boundaries & non-compact · Shared terms “isometry”

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) Scalar curvature The trace of the Ricci tensor. Under Ricci flow its minimum never decreases and positive scalar curvature (PSC) is preserved. Start with: On the proof of the positive mass conjecture in general relativity (Schoen et al., 1979) 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)

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. 6C.4
    The Yamabe Problem and Critical ExponentsBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Conformal changes of scalar curvature, the Yamabe functional, Aubin's threshold and Schoen's solution, concentration compactness and bubbles, and why Perelman's log-Sobolev minimisers always exist.

  3. 9B.6
    Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds

    Positive scalar curvature, its obstructions, and the positive mass theorem.