Paper · Jul 30, 2024
Topological Obstructions to the Existence of Compact Shrinking Ricci Solitons in Dimension Four
As on arXiv
Abstract
This undergraduate thesis is focused on introducing the reader to concepts related to the search for topological obstructions to the existence of compact gradient shrinking Ricci soliton metrics in dimension four. It contains a discussion of the relevant background material for this subject. Furthermore, it introduces the problem of extending the Hitchin-Thorpe inequality to gradient shrinking Ricci soliton metrics and explores the limitations of current results in that direction. At last, the topic of compact Kaehler gradient shrinking Ricci solitons is introduced and the classification of these spaces is outlined in literature-study fashion.
- Submitted
- Jul 30, 2024
- Last revised
- Jul 24, 2026 (v3)
- Subjects
- math.DG
Comments: 18 pages, Undergraduate Thesis. This is a second version. The Author was not aware of reference 1, which proved theorem 4.4 independently of Cao, to whom the theorem was credited in version 1. See also the "Editor's Note" for some corrections to Section 5, contributed by Brian Klatt, for which the author is grateful
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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
- 11B.1Ricci Solitons
Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.
- 12B.2The Asymptotic Soliton and Three-Dimensional Shrinkers
Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.