As on arXiv
Abstract
We consider noncollapsed steady gradient Ricci solitons with nonnegative sectional curvature. We show that such solitons always dimension reduce at infinity. This generalizes an earlier result in [CDM22] to higher dimensions. In dimension four, we classify possible reductions at infinity, which lays foundation for possible classifications of steady solitons. Moreover, we show that any tangent flow at infinity of a general noncollapsed steady soliton must split off a line. This generalizes an earlier result in [BCDMZ21] to higher dimensions. While this article is under preparation, we realized that part of our main results are proved independently in a recent post [ZZ23] under different assumptions.
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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.
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An optimal volume growth estimate for noncollapsed steady gradient Ricci solitons
Topics (Non)collapsing, Ricci solitons · Same authors Pak-Yeung Chan, Zilu Ma, Yongjia Zhang · Shared terms “noncollapsed steady”, “steady gradient”, “gradient Ricci”
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On noncollapsed -limit metric solitons
Topics (Non)collapsing, Singularities, Ricci solitons · Same authors Pak-Yeung Chan, Zilu Ma, Yongjia Zhang · Shared terms “noncollapsed”
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Rigidity of positively curved Steady gradient Ricci solitons on orbifolds
Topics (Non)collapsing, Singularities, Ricci solitons · Shared terms “curved steady”, “positively curved”, “noncollapsed steady”
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steady gradient Ricci solitons with nonnegative curvature away from a compact set
Topics 4-manifolds, (Non)collapsing, Ricci solitons · Shared terms “noncollapsed steady”, “steady gradient”, “nonnegative sectional”
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Correction to: Curvature growth of some 4-dimensional gradient Ricci soliton singularity models
Topics 4-manifolds, Singularities, Ricci solitons · Same author Yongjia Zhang · Shared terms “gradient Ricci”
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On the fundamental group of steady gradient Ricci solitons with nonnegative sectional curvature
Topics (Non)collapsing, Ricci solitons · Shared terms “nonnegative sectional”, “noncollapsed steady”, “steady gradient”
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Hamilton-Ivey estimates for gradient Ricci solitons
Topics Singularities, Ricci solitons · Same authors Pak-Yeung Chan, Zilu Ma, Yongjia Zhang · Shared terms “steady gradient”, “gradient Ricci”
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On a dichotomy of the curvature decay of steady Ricci soliton
Topics 4-manifolds, Ricci solitons · Same author Pak-Yeung Chan · Shared terms “higher dimensions”, “steady gradient”, “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.
In the guidebook
- 9B.3Collapsing and Noncollapsing
Injectivity radius bounds from volume, collapsing examples, and the definition Perelman needs.
- 12A.4κ-Noncollapsing
The theorem that excludes the cigar.
- 12A.8Local Noncollapsing from the Reduced Volume
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.