Paper · Nov 18, 2021
Curvature estimates for four-dimensional complete gradient expanding Ricci solitons
As on arXiv
Abstract
In this paper, we derive curvature estimates for 4-dimensional complete gradient expanding Ricci solitons with nonnegative Ricci curvature (outside a compact set ). More precisely, we prove that the norm of the curvature tensor and its covariant derivative can be bounded by the scalar curvature by and (on ), for any and some constant . Moreover, if the scalar curvature has at most polynomial decay at infinity, then (on ). As an application, it follows that that if a 4-dimensional complete gradient expanding Ricci soliton has nonnegative Ricci curvature and finite asymptotic scalar curvature ratio then it has finite asymptotic curvature ratio, and asymptotic cones at infinity () according to Chen-Deruelle [20].
Comments: 20 pages; to appear in Crelle's Journal
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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.
- 11B.1Ricci Solitons
Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.