Paper · Jun 6, 2016
Structure at infinity for shrinking Ricci solitons
As on arXiv
Abstract
The paper mainly concerns the structure at infinity for complete gradient shrinking Ricci solitons. It is shown that for such a soliton with bounded curvature, if the round cylinder occurs as a limit for a sequence of points going to infinity along an end, then the end is asymptotic to the same round cylinder at infinity. The result is then applied to obtain structural results at infinity for four dimensional gradient shrinking Ricci solitons. It is previously known that such solitons with scalar curvature approaching zero at infinity must be smoothly asymptotic to a cone. For the case that the scalar curvature is bounded from below by a positive constant, we conclude that along each end the soliton is asymptotic to a quotient of or converges to a quotient of along each integral curve of the gradient vector field of the potential function.
- Submitted
- Jun 6, 2016
- Last revised
- Apr 8, 2022 (v2)
- Subjects
- math.DG, math.AP
Comments: Published version
Its neighbourhood
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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.
Foundational paper in the Ricci solitons guide: Munteanu–Wang: the ends of four-dimensional shrinkers with bounded scalar curvature.
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.