As on arXiv
Abstract
We study blow-up sequences of Ricci shrinkers without global curvature assumptions based at points at which the scalar curvature satisfies a Type I bound, proving that their -limits split a line. In the four-dimensional case these limits are smooth Ricci shrinkers and the convergence is in the pointed smooth Cheeger-Gromov sense. As a consequence, limits along the integral curve of starting at such a point split a line. This generalises known results about the geometry of ends of Ricci shrinkers that relied on global curvature bounds. To obtain our results, we extend the -convergence theory from Bamler and Li-Wang.
Comments: 47 pages. All comments welcome
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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.
In the guidebook
- 6C.4The Yamabe Problem and Critical Exponents
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.6Scalar Curvature and Topology
Positive scalar curvature, its obstructions, and the positive mass theorem.
- 6A.13Self-Similar Variables and Monotonicity
Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.