Paper · Oct 24, 2022

Geometric Flows of -Structures on 3-Sasakian 7-Manifolds

Aaron Kennon, Jason D. Lotay

math.DGarXiv:2210.12962v2J Geom Phys 187 (2023), 104793
PDF Abstract page

As on arXiv

Abstract

A 3-Sasakian structure on a 7-manifold may be used to define two distinct Einstein metrics: the 3-Sasakian metric and the squashed Einstein metric. Both metrics are induced by nearly parallel -structures which may also be expressed in terms of the 3-Sasakian structure. Just as Einstein metrics are critical points for the Ricci flow up to rescaling, nearly parallel -structures provide natural critical points of the (rescaled) geometric flows of -structures known as the Laplacian flow and Laplacian coflow. We study each of these flows in the 3-Sasakian setting and see that their behaviour is markedly different, particularly regarding the stability of the nearly parallel -structures. We also compare the behaviour of the flows of -structures with the (rescaled) Ricci flow.

Comments: 23 pages and 3 figures; v2: minor changes, version accepted for publication in Journal of Geometry and Physics

Ricci flowGeneralized & G₂ flowsEinstein metricsStability & uniquenessSasakian & contact geometry

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 Geometry of para-Sasakian metric as an almost conformal η-Ricci soliton (Sumanjit Sarkar, Santu Dey and Arindam Bhattacharyya, 2021)5Sarkar 2021 Convergence of the Sasaki-Ricci flow on Sasakian 5-manifolds of general type (Shu-Cheng Chang, Yingbo Han, Chien Lin et al., 2022)7Chang 2022 On the Generalised Ricci Solitons and Sasakian Manifolds (Ahmed Mohammed Cherif, Kaddour Zegga and Gherici Beldjilali, 2022)1Cherif 2022 Generalized Ricci solitons and Einstein metrics on weak K-contact manifolds (Vladimir Rovenski, 2023)3Rovenski 2023 The canonical symmetry reduction of string backgrounds (Aaron Kennon and Jeffrey Streets, 2025)2Kennon 2025 Ricci solitons as critical points of quadratic curvature functionals (Atreyee Bhattacharya and Sayoojya Prakash, 2026)8Bhattacharya 2026 Sasaki with torsion manifolds and string backgrounds (Beatrice Brienza, Anna Fino, Udhav Fowdar et al., 2026)6Brienza 2026 Characterization of Sasakian Manifolds by Curvature Conditions (Sourav Nayak, Vladimir Rovenski and Dhriti Sundar Patra, 2026)4Nayak 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On the Generalised Ricci Solitons and Sasakian Manifolds

    Ahmed Mohammed Cherif, Kaddour Zegga and Gherici Beldjilali PDF

    Topics Sasakian & contact geometry, Generalized & G₂ flows · Shared terms “Sasakian structure”

  2. The canonical symmetry reduction of string backgrounds

    Aaron Kennon and Jeffrey Streets PDF

    Topics Sasakian & contact geometry, Generalized & G₂ flows, Stability & uniqueness · Same author Aaron Kennon

  3. Generalized Ricci solitons and Einstein metrics on weak -contact manifolds

    Vladimir Rovenski PDF

    Topics Sasakian & contact geometry, Generalized & G₂ flows, Einstein metrics · Shared terms “parallel”

  4. Characterization of Sasakian Manifolds by Curvature Conditions

    Sourav Nayak, Vladimir Rovenski and Dhriti Sundar Patra PDF

    Topics Sasakian & contact geometry · Shared terms “Sasakian structure”

  5. Geometry of para-Sasakian metric as an almost conformal -Ricci soliton

    Sumanjit Sarkar, Santu Dey and Arindam Bhattacharyya PDF

    Topics Sasakian & contact geometry, Einstein metrics

  6. Sasaki with torsion manifolds and string backgrounds

    Beatrice Brienza, Anna Fino, Udhav Fowdar et al. PDF

    Topics Sasakian & contact geometry, Generalized & G₂ flows, Einstein metrics · Shared terms “geometric flows”

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

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

    Topics Sasakian & contact geometry, Einstein metrics

  8. Ricci solitons as critical points of quadratic curvature functionals

    Atreyee Bhattacharya and Sayoojya Prakash PDF

    Topics Einstein metrics, Stability & uniqueness · Shared terms “critical points”

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) Generalized & G₂ flows Ricci-type flows for geometry beyond Kähler metrics. Generalized Ricci flow couples a metric to a closed 3-form, pluriclosed and Chern–Ricci flows… Start with: A parabolic flow of pluriclosed metrics (Streets et al., 2009) 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) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005)

In the guidebook

  1. 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.

  2. 10A.12
    Curve Shortening and Minimal DiscsBook 10A · Three-Manifolds and Geometrization

    How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.

  3. 11A.1
    The Equation and Its First SolutionsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A heat equation for the metric: spheres, cylinders and scaling.