Paper · Mar 21, 2024

Transverse Ricci solitons on a compact foliated manifold

Seungsu Hwang, Seoung Dal Jung, Jungwoo Moon

math.DGarXiv:2403.14099
View PDF View HTML arXiv page

As on arXiv

Abstract

We investigate transverse Ricci solitons, the self-similar solutions of the transverse Ricci flow, on a compact foliated manifold. In particular, we show the relations between a taut Riemannian foliation and a transverse Ricci soliton. Moreover, we find some examples of transverse Ricci solitons.

View PDF View HTML arXiv page
Submitted
Mar 21, 2024
Subjects
math.DG

Comments: 32 pages

Ricci flowRicci solitons

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 Convergence of the Sasaki-Ricci flow on Sasakian 5-manifolds of general type (Shu-Cheng Chang, Yingbo Han, Chien Lin et al., 2022)3Chang 2022 Vaisman manifolds and transversally Kähler-Einstein metrics (Vladimir Slesar and Gabriel-Eduard Vîlcu, 2022)6Slesar 2022 Gradient Shrinking Sasaki-Ricci Solitons on Sasakian Manifolds of Dimension Up to Seven (Der-Chen Chang, Shu-Cheng Chang, Yingbo Han et al., 2022)2Chang 2022 Translators of the Mean Curvature Flow in Hyperbolic Einstein's Static Universe (Miguel Ortega and Buse Yalçın, 2024)7Ortega 2024 On the existence of conic Sasaki-Einstein metrics on log Fano Sasakian manifolds of dimension five (Shu-Cheng Chang, Fengjiang Li, Chien Lin et al., 2024)8Chang 2024 Transverse Rigidity of Shrinking Sasaki-Ricci Solitons (Shu-Cheng Chang, Fengjiang Li, Chien Lin et al., 2025)4Chang 2025 On the Hamilton-Tian Conjecture in a compact transverse Fano Sasakian 5-manifold (Shu-Cheng Chang, Yingbo Han, Chien Lin et al., 2026)5Chang 2026 On a compact Einstein-type manifolds with Riemannian foliations (Jungwoo Moon, 2026)1Moon 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On a compact Einstein-type manifolds with Riemannian foliations

    Jungwoo Moon

    Topics Ricci solitons · Same author Jungwoo Moon · Shared terms “Riemannian foliation”, “taut Riemannian”

  2. Gradient Shrinking Sasaki-Ricci Solitons on Sasakian Manifolds of Dimension Up to Seven

    Der-Chen Chang, Shu-Cheng Chang, Yingbo Han et al.

    Topics Ricci solitons · Shared terms “transverse Ricci”, “foliation”

  3. Convergence of the Sasaki-Ricci flow on Sasakian 5-manifolds of general type

    Shu-Cheng Chang, Yingbo Han, Chien Lin et al.

    Topics Ricci flow · Shared terms “transverse Ricci”, “foliation”

  4. Transverse Rigidity of Shrinking Sasaki-Ricci Solitons

    Shu-Cheng Chang, Fengjiang Li, Chien Lin et al.

    Topics Ricci solitons · Shared terms “transverse”

  5. On the Hamilton-Tian Conjecture in a compact transverse Fano Sasakian -manifold

    Shu-Cheng Chang, Yingbo Han, Chien Lin et al.

    Topics Ricci solitons · Shared terms “transverse”, “foliation”

  6. Vaisman manifolds and transversally Kähler-Einstein metrics

    Vladimir Slesar and Gabriel-Eduard Vîlcu

    Topics Ricci flow · Shared terms “transverse”, “foliation”

  7. Translators of the Mean Curvature Flow in Hyperbolic Einstein's Static Universe

    Miguel Ortega and Buse YalçınWider flows

    Shared terms “foliated”

  8. On the existence of conic Sasaki-Einstein metrics on log Fano Sasakian manifolds of dimension five

    Shu-Cheng Chang, Fengjiang Li, Chien Lin et al.

    Topics Ricci solitons · Shared terms “transverse”, “foliation”

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 flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) 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)

In the guidebook

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

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

  2. 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.

  3. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.