Topological Obstructions to the Existence of Compact Shrinking Ricci Solitons in Dimension Four

TL;DR

This thesis explores topological obstructions to the existence of compact gradient shrinking Ricci solitons in four dimensions, discussing background, extending inequalities, and classifying Kähler cases.

Contribution

It introduces the problem of extending the Hitchin-Thorpe inequality to Ricci solitons and reviews classification results for compact Kähler shrinking Ricci solitons.

Findings

Discussion of topological obstructions in 4D Ricci solitons

Analysis of limitations in extending Hitchin-Thorpe inequality

Outline of classification of compact Kähler Ricci solitons

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.