Geometric flows of G2-structures, and their Yang-Mills connections.

• Grant number: 24/13414-2
• Support Opportunities: Scholarships abroad - Research Internship - Post-doctor
• Start date: November 01, 2024
• End date: June 30, 2025
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Henrique Nogueira de Sá Earp
• Grantee: Jacob Richard Stein
• Supervisor: Jason Lotay
• Host Institution: Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil
• Institution abroad: University of Oxford, England
• Associated to the scholarship: 23/02809-3 - Singular G2-geometry, gauge theory and co-dimension one collapse, BP.PD

Abstract

This proposal studies solutions of two geometric flows, one of G2-structures, and one of Hermitian metrics on vector bundles, in order to further the study of G2-manifolds and their gauge theory. Thefirst part, building on the examples in the candidate's previous work, will be to establish an elliptic deformationtheory for solutions of the so-called Hermitian-Yang-Mills heat-flow over asymptotically conical Calabi-Yau3-folds, and prove suitable decay estimates for these solutions. This is motivated by a proposed constructionof gauge-theoretic objects called G2-monopoles on non-compact manifolds, currently known to exist in onlya handful of examples. The second part of this project will be to investigate the dynamic stability of an important class of G2-structures called nearly parallel, under the modified Laplacian co-flow. While this flow isa promising candidate to produce special metrics, particularly in the examples currently being studied by thecandidate, there are key open questions regarding its long-time behaviour.