Paper · Apr 24, 2016

Ridigity of Ricci Solitons with Weakly Harmonic Weyl Tensors

Seungsu Hwang, Gabjin Yun

math.DGarXiv:1604.07018
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Abstract

In this paper, we prove rigidity results on gradient shrinking Ricci solitons with weakly harmonic Weyl curvature tensors. Let be a compact gradient shrinking Ricci soliton satisfying with constant. We show that if satisfies , then is Einstein. Here denotes the Weyl curvature tensor. In the case of noncompact, if is complete and satisfies the same condition, then is rigid in the sense that is given by a quotient of product of an Einstein manifold with Euclidean space. These are generalizations of the previous known results in, and.

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Submitted
Apr 24, 2016
Subjects
math.DG

Comments: 15pages

Ricci solitonsEinstein metricsStability & uniquenessBoundaries & non-compact

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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. Bach-flat h-almost gradient Ricci solitons

    Gabjin Yun, Jinseok Co and Seungsu Hwang

    Topics Boundaries & non-compact, Ricci solitons · Same authors Seungsu Hwang, Gabjin Yun · Shared terms “harmonic Weyl”, “Weyl curvature”

  2. Rigidity of Complete Gradient Steady Ricci Solitons with Harmonic Weyl Curvature

    Fengjiang Li

    Topics Boundaries & non-compact, Stability & uniqueness, Ricci solitons · Shared terms “harmonic Weyl”, “Weyl curvature”, “Weyl tensor”

  3. On a classification of 4-d gradient Ricci solitons with harmonic Weyl curvature

    Jongsu Kim

    Topics Einstein metrics, Stability & uniqueness, Ricci solitons · Shared terms “harmonic Weyl”, “Weyl curvature”, “Euclidean”

  4. Gradient Shrinking Sasaki-Ricci Solitons with Harmonic Weyl Tensor

    Shu-Cheng Chang and Hongbing Qiu

    Topics Ricci solitons · Shared terms “harmonic Weyl”, “Weyl tensor”, “gradient shrinking”

  5. Classification of Gradient Ricci solitons with harmonic Weyl curvature

    Jongsu Kim

    Topics Einstein metrics, Ricci solitons · Shared terms “harmonic Weyl”, “Weyl curvature”

  6. Gradient Ricci solitons with vanishing conditions on Weyl

    Giovanni Catino, Paolo Mastrolia and Dario Daniele Monticelli

    Topics Einstein metrics, Ricci solitons · Shared terms “Weyl tensor”, “soliton satisfying”, “gradient shrinking”

  7. The Ricci tensor of a gradient Ricci soliton with harmonic Weyl tensor

    Valter Borges, Matheus Andrade Ribeiro de Moura Horácio and João Paulo dos Santos

    Topics Ricci solitons · Shared terms “harmonic Weyl”, “Weyl tensor”

  8. Complete gradient Ricci solitons with zero radial Weyl curvature

    Tongzhu Li and Junlong Yu

    Topics Ricci solitons · Shared terms “Weyl curvature”, “Weyl tensor”

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) 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) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989)

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.