Paper · Oct 19, 2016

On the existence of stationary Ricci solitons

Pau Figueras, Toby Wiseman

gr-qcarXiv:1610.06178
View PDF View HTML arXiv page

As on arXiv

Abstract

Previously the DeTurck 'trick' has been used to render the stationary Einstein's equation a well posed elliptic system that may be solved numerically by geometric flow or directly. Whilst in the static case for pure gravity with zero or negative cosmological constant there is a simple proof that solving the modified "harmonic" Einstein's equation leads to a solution of the original Einstein system - i.e. not a Ricci soliton - in the stationary case this argument no longer works. Here we provide a new argument that extends the static result to the case of stationary spacetimes that possess a "-" reflection symmetry. Defining a "soliton charge" from the asymptotic behaviour of the solution, we show that this quantity is always non-positive. Provided asymptotic conditions are chosen such that this charge vanishes, then stationary solitons cannot exist.

View PDF View HTML arXiv page
Submitted
Oct 19, 2016
Subjects
gr-qc, hep-th
DOI
10.1088/1361-6382/aa764a

Comments: 15 pages, no figures

Ricci solitonsHarmonic maps & DeTurck trickPhysicsNumerics & visualization

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 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 Some geometrical properties of the Oscillator group (Yadollah Aryanejad, 2016)8Aryanejad 2016 Topological Quantum Gravity of the Ricci Flow (Alexander Frenkel, Petr Horava and Stephen Randall, 2020)3Frenkel 2020 Perelman's Ricci Flow in Topological Quantum Gravity (Alexander Frenkel, Petr Horava and Stephen Randall, 2020)4Frenkel 2020 Gradient flow of Einstein-Maxwell theory and Reissner-Nordström black holes (Davide De Biasio, Julian Freigang, Dieter Lust et al., 2022)1De Biasio 2022 A Statistical Fields Theory underlying the Thermodynamics of Ricci Flow and Gravity (M. J. Luo, 2023)6Luo 2023 Error analysis of a finite element scheme for parametric mean curvature flow based on the DeTurck trick (Klaus Deckelnick and Vanessa Styles, 2026)5Deckelnick 2026 Noise-induced stabilization of Schwarzschild–AdS black holes under stochastic Ricci flow (Jihui Wang, Matteo Lulli and Antonino Marciano, 2026)7Wang 2026 Using Ricci Flow in the Construction of Homogeneous Cosmological Models (Fan Zhang and Lee Lindblom, 2026)2Zhang 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Gradient flow of Einstein-Maxwell theory and Reissner-Nordström black holes

    Davide De Biasio, Julian Freigang, Dieter Lust et al.

    Topics Numerics & visualization, Physics · Same author Toby Wiseman · Shared terms “numerically”

  2. Using Ricci Flow in the Construction of Homogeneous Cosmological Models

    Fan Zhang and Lee Lindblom

    Topics Numerics & visualization, Physics · Shared terms “Einstein equations”, “cosmological”, “numerically”

  3. Topological Quantum Gravity of the Ricci Flow

    Alexander Frenkel, Petr Horava and Stephen Randall

    Topics Harmonic maps & DeTurck trick, Physics

  4. Perelman's Ricci Flow in Topological Quantum Gravity

    Alexander Frenkel, Petr Horava and Stephen Randall

    Topics Harmonic maps & DeTurck trick, Physics, Ricci solitons

  5. Error analysis of a finite element scheme for parametric mean curvature flow based on the DeTurck trick

    Klaus Deckelnick and Vanessa StylesWider flows

    Topics Numerics & visualization, Harmonic maps & DeTurck trick

  6. A Statistical Fields Theory underlying the Thermodynamics of Ricci Flow and Gravity

    M. J. Luo

    Topics Harmonic maps & DeTurck trick, Physics, Ricci solitons · Shared terms “cosmological constant”

  7. Noise-induced stabilization of Schwarzschild–AdS black holes under stochastic Ricci flow

    Jihui Wang, Matteo Lulli and Antonino Marciano

    Topics Numerics & visualization, Physics · Shared terms “negative cosmological”, “cosmological constant”

  8. Some geometrical properties of the Oscillator group

    Yadollah Aryanejad

    Topics Harmonic maps & DeTurck trick, Physics, Ricci solitons · Shared terms “defining”

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) Harmonic maps & DeTurck trick Ricci flow is only weakly parabolic, because it is invariant under diffeomorphisms. The DeTurck trick adds a Lie-derivative term that makes it… Start with: Harmonic Mappings of Riemannian Manifolds (Eells et al., 1964) Physics Ricci flow first appeared in physics, as the leading-order renormalization-group flow of the string sigma model (Friedan, 1980). Start with: Nonlinear Models in 2+ε Dimensions (Friedan, 1980) Numerics & visualization Computing and visualising Ricci flow: simulations of neckpinches and critical behaviour, pictures of rotationally symmetric flows, and… Start with: Deforming metrics in the direction of their Ricci tensors (DeTurck, 1983)

In the guidebook

  1. 6D.11
    Computing Geometric FlowsBook 6D · Diffusions, Finance and Computation

    Numerical curve shortening and mean curvature flow, level sets and phase fields, discrete Ricci flow on surfaces and in conformal mapping, rotationally symmetric neckpinches, and what computation has taught about singularities.

  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.