Cameron MacMahon | Searcharxiv

arXiv2026

Convex Hull Volumes in Hyperbolic 3-Space

In this paper we provide a geometric condition satisfied by certain closed subsets of the Riemann sphere which implies that their hyperbolic convex hulls in $\mathbb{H}^3$ have infinite volume. As a corollary, we characterize continua in the Riemann sphere whose hyperbolic convex hulls have infinite volume, answering a question of Danny Calegari. Furthermore, we give a geometric characterization of planar self-similar sets whose hyperbolic convex hulls have infinite volume.

math.GT↗

arXiv2024

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

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.

math.DG↗