Steady Ricci solitons with horizontally ε-pinched Ricci curvature

Yuxing Deng; Xiaohua Zhu

Paper · Jan 9, 2016

Steady Ricci solitons with horizontally $ε$ -pinched Ricci curvature

Yuxing Deng, Xiaohua Zhu

math.DGarXiv:1601.02111v3

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Abstract

In this paper, we prove that any $κ$ -noncollapsed gradient steady Ricci soliton with nonnegative curvature operator and horizontally $ε$ -pinched Ricci curvature must be rotationally symmetric. As an application, we show that any $κ$ -noncollapsed gradient steady Ricci soliton $(M^{n}, g, f)$ with nonnegative curvature operator must be rotationally symmetric if it admits a unique equilibrium point and its scalar curvature $R (x)$ satisfies $lim_{r (x) \to \infty} R (x) f (x) = C_{0} sup_{x \in M} R (x)$ with $C_{0} > \frac{n - 2}{2}$ .

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Submitted: Jan 9, 2016
Last revised: Dec 4, 2016 (v3)
Subjects: math.DG

Comments: Corollary 3.11 is added

Ricci solitons (Non)collapsing Scalar curvature Rotational symmetry & warped products

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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) (Non)collapsing At scales where curvature is controlled, balls in a Ricci flow cannot be arbitrarily thin: Perelman's κ-noncollapsing theorem. Start with: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds (Cheeger et al., 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) Rotational symmetry & warped products For rotationally symmetric metrics and warped products, Ricci flow reduces to a parabolic equation in one space variable. Start with: The ricci flow on radially symmetric R3 (Ivey, 1994)

In the guidebook

9B.3
Collapsing and NoncollapsingBook 9B · Comparison, Convergence and Heat on Manifolds
Injectivity radius bounds from volume, collapsing examples, and the definition Perelman needs.
12A.4
κ-NoncollapsingBook 12A · Entropy and Noncollapsing
The theorem that excludes the cigar.
12A.8
Local Noncollapsing from the Reduced VolumeBook 12A · Entropy and Noncollapsing
Perelman's §§7.3 and 8: noncollapsing that needs only local curvature control, the small-and-large-vector split, the maximum principle for the reduced distance, and the form that survives surgery.

[1] 9B.3
Collapsing and NoncollapsingBook 9B · Comparison, Convergence and Heat on Manifolds
Injectivity radius bounds from volume, collapsing examples, and the definition Perelman needs.

[2] 12A.4
κ-NoncollapsingBook 12A · Entropy and Noncollapsing
The theorem that excludes the cigar.

[3] 12A.8
Local Noncollapsing from the Reduced VolumeBook 12A · Entropy and Noncollapsing
Perelman's §§7.3 and 8: noncollapsing that needs only local curvature control, the small-and-large-vector split, the maximum principle for the reduced distance, and the form that survives surgery.

Steady Ricci solitons with horizontally $ε$ -pinched Ricci curvature

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