Rigidity of four-dimensional Gradient shrinking Ricci solitons

Xu Cheng; Detang Zhou

Paper · May 22, 2021

Rigidity of four-dimensional Gradient shrinking Ricci solitons

Xu Cheng, Detang Zhou

math.DGarXiv:2105.10744v2

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Abstract

Let $(M, g, f)$ be a $4$ -dimensional complete noncompact gradient shrinking Ricci soliton with the equation $R i c + \nabla^{2} f = λ g$ , where $λ$ is a positive real number. We prove that if $M$ has constant scalar curvature $S = 2 λ$ , it must be a quotient of $S^{2} \times R^{2}$ . Together with the known results, this implies that a $4$ -dimensional complete gradient shrinking Ricci soliton has constant scalar curvature if and only if it is rigid, that is, it is either Einstein, or a finite quotient of Gaussian shrinking soliton $R^{4}$ , $S^{2} \times R^{2}$ or $S^{3} \times R$ .

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Submitted: May 22, 2021
Last revised: Jun 22, 2021 (v2)
Subjects: math.DG

Comments: 22 pages. Comments are welcome

Ricci solitons 4-manifolds Scalar curvature Stability & uniqueness Boundaries & non-compact

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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) 4-manifolds Four is the first dimension where the singularities of Ricci flow are not classified. It is also the dimension of the still-open smooth Poincaré… Start with: The topology of four-dimensional manifolds (Freedman, 1982) Scalar curvature The trace of the Ricci tensor. Under Ricci flow its minimum never decreases and positive scalar curvature (PSC) is preserved. Start with: On the proof of the positive mass conjecture in general relativity (Schoen et al., 1979) 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

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.
9B.6
Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds
Positive scalar curvature, its obstructions, and the positive mass theorem.
11B.1
Ricci SolitonsBook 11B · Solitons, Compactness and Singularities
Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.

[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.

Rigidity of four-dimensional Gradient shrinking Ricci solitons

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