Topological Obstructions to the Existence of Compact Shrinking Ricci Solitons in Dimension Four
Cameron MacMahon
TL;DR
The thesis investigates topological obstructions to the existence of compact gradient shrinking Ricci solitons in four dimensions, aiming to extend Hitchin–Thorpe type inequalities to shrinking solitons and to exploit four‑manifold invariants. It derives two sufficient conditions that guarantee the Hitchin–Thorpe inequality for shrinking solitons and develops a conformal‑geometric framework involving the Yamabe invariant and the invariant $\int \sigma_2(A_g) \, dV_g$ to isolate a class of solitons for which the obstruction holds, while acknowledging limitations evidenced by known non‑Einstein examples. In the Kähler setting, the work leverages positivity of the first Chern class to identify del‑Pezzo surfaces and achieves a complete classification: the compact Kähler gradient shrinking Ricci solitons are exactly the Kähler–Einstein examples on del‑Pezzo surfaces together with the Koiso–Cao and Wang–Zhu non‑Einstein solitons, up to holomorphic automorphisms. The study clarifies where topological obstructions constrain shrinking solitons and highlights open questions for non‑Kähler and broader geometric structures, emphasizing the roles of $b_1$, $χ$, $τ$, and curvature decompositions in four dimensions.
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.