Paper · Oct 27, 2021

Gradient Ricci solitons carrying a closed conformal vector field

J. F. Siva Filho, R. Sharma

math.DGarXiv:2110.14103
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Abstract

We show that a complete gradient Ricci soliton with constant scalar curvature and a non-parallel closed conformal vector field is isometric to either the Euclidean space, or an Euclidean sphere, or negatively Einstein warped product of the real line with a complete non-positively Einstein manifold. Moreover, we show that a Kähler gradient Ricci soliton of real dimension , with a non-parallel closed conformal real vector field is Ricci-flat (Calabi-Yau) and is flat in dimension 4. Finally, we show that a Ricci soliton whose associated 1-form is harmonic, has constant scalar curvature.

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Submitted
Oct 27, 2021
Subjects
math.DG

Comments: 10 pages

Ricci solitonsEinstein metricsRicci-flat & Calabi–Yau4-manifoldsKähler geometryScalar curvatureRotational symmetry & warped products

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  1. Classification of Gradient Ricci solitons with harmonic Weyl curvature

    Jongsu Kim

    Topics Ricci-flat & Calabi–Yau, Rotational symmetry & warped products, 4-manifolds · Shared terms “gradient Ricci”, “complete gradient”

  2. New expanding Ricci solitons starting in dimension four

    Jan Nienhaus and Matthias Wink

    Topics Ricci-flat & Calabi–Yau, 4-manifolds, Einstein metrics

  3. On Ricci solitons whose level hypersurfaces have parallel second fundamental form

    Matheus Andrade Ribeiro de Moura Horácio

    Topics Rotational symmetry & warped products, Scalar curvature, Ricci solitons · Shared terms “parallel”, “gradient Ricci”, “constant scalar”

  4. Rigidity of nonsteady gradient Kähler-Ricci solitons with constant scalar curvature

    Matheus Andrade Ribeiro de Moura Horácio

    Topics Scalar curvature, Kähler geometry, Ricci solitons · Shared terms “constant scalar”, “vector field”, “gradient Ricci”

  5. On the geometry of Riemannian warped product maps

    Jyoti Yadav, Harmandeep Kaur and Gauree Shanker

    Topics Rotational symmetry & warped products, Scalar curvature, Ricci solitons

  6. Kähler–Ricci Shrinkers with Constant Scalar Curvature Are Rigid

    Wenqi Li, Jianyu Ou, Guoqiang Wu et al.

    Topics Einstein metrics, Scalar curvature, Kähler geometry · Shared terms “constant scalar”, “gradient Ricci”, “complete gradient”

  7. On some locally symmetric embedded spaces with non-negative scalar curvature and their characterization

    Abbas M Sherif, Peter K S Dunsby and Rituparno Goswami

    Topics Rotational symmetry & warped products, Scalar curvature, Ricci solitons · Shared terms “vector field”

  8. A note on four dimensional Shrinking Gradient Ricci Solitons with Constant Scalar Curvature

    Chen Wang and Guoqiang Wu

    Topics 4-manifolds, Scalar curvature, Ricci solitons · Shared terms “constant scalar”, “gradient 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) 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) Ricci-flat & Calabi–Yau Ricci-flat metrics do not move under Ricci flow. Every known compact simply connected example has special holonomy (Calabi–Yau, hyperkähler, G₂… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) 4-manifolds Four is the first dimension where Ricci flow's singularities are not fully understood. It is also the dimension of the smooth Poincaré conjecture… Start with: The topology of four-dimensional manifolds (Freedman, 1982)

In the guidebook

  1. 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.

  2. 9B.6
    Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds

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

  3. 11B.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

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