Open Collections

Suan, Caleb

University of British Columbia

In this thesis, we study three problems in non-Kähler Calabi–Yau geometry and G₂-geometry centered on Reid’s Fantasy [Rei87] and the Hull–Strominger system [Hul86, Str86]. The first concerns the geometrization of conifold transitions, processes which allow us to traverse the moduli space of compact Calabi–Yau threefolds. Work of Fu–Li–Yau [FLY12] and of Collins–Picard–Yau [CPY24] has constructed metrics on both sides of this process which are partial solutions to the Hull–Strominger system. Using these (conformally) balanced and Hermitian Yang–Mills metrics, we show that conifold transitions are continuous in the Gromov–Hausdorff topology. The next focuses on the Anomaly flow of Phong–Picard–Zhang [PPZ18b]. We extend their ideas from the 𝛼′ = 0 case and compute integral Shi-type estimates along the flow for general slope parameter 𝛼′. We achieve this by adapting an integration-by-parts type argument instead of the usual Maximum Principle techniques in order to deal with the extra terms that appear. From this, we obtain a smallness condition on 𝛼′ that allows the flow to be extended from [0,τ) to a larger interval [0,τ + ϵ). Finally, we study the relationship between Calabi–Yau geometry and G₂-geometry by considering geometric flows on S¹-fibrations over Calabi–Yau threefolds. In particular, we construct families of closed and coclosed G₂-structures on these fibrations and apply the Laplacian flow and (modified) coflow respectively. Using these Ans¨atze, we show that on a trivial fibration these flows reduce to particular Monge–Ampère flows on the base manifold. We perform a similar analysis on contact Calabi–Yau 7-folds and obtain conditions for these families to satisfy the flows.

Text

2025-04-08

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/90600

Mathematics

University of British Columbia

UBCV

http://creativecommons.org/licenses/by-nc-nd/4.0/