Shubham Dwivedi, Department of Pure Mathematics, University of Waterloo
"Topics in G_2 geometry and geometric flows"
We study three different problems in this thesis, all related to G_2 structures and geometric flows. In the first problem we study hypersurfaces in a nearly G_2 manifold. We give a necessary and sufficient condition for a hypersurface with an almost complex structure induced from the G_2 structure of the ambient manifold to be nearly Kähler. Then using the nearly G_2 structure on the round sphere S^7, we prove that for a compact minimal hypersurface M^6 of constant scalar curvature in S^7 with the shape operator A satisfying |A|^2 > 6, there exists an eigenvalue λ > 12 of the Laplace operator on M such that |A|^2 = λ − 6, thus giving the next discrete value of |A|^2 greater than 0 and 6. The latter is related to a question of Chern on the values of the scalar curvature of compact minimal hypersurfaces in S^n of constant scalar curvature.
The second problem is related to the study of solitons and almost solitons of the Ricci- Bourguignon flow. We prove some characterization results for compact Ricci-Bourguignon solitons and almost solitons.
In the third problem we study a flow of G_2 structures that all induce the same Riemannian metric. This isometric flow is the negative gradient flow of an energy functional. We prove Shi-type estimates for the torsion tensor T along the flow. We show that at a finite-time singularity the torsion must blow up, so the flow will exist as long as the torsion remains bounded. We prove a Cheeger–Gromov type compactness theorem for the flow. We define a scale-invariant quantity for any solution of the flow and the proof that it is almost monotonic along the flow. We also introduce an entropy functional and prove that low entropy initial data lead to solutions of the flow that exist for all time and converge smoothly to a G_2 structure with divergence-free torsion. We study the singular set of the flow. Finally, we prove that if the singularity is of Type-I then a sequence of blow-ups of a solution admits a subsequence that converges to a shrinking soliton for the flow.