## Contact Info

Pure MathematicsUniversity of Waterloo

200 University Avenue West

Waterloo, Ontario, Canada

N2L 3G1

Departmental office: MC 5304

Phone: 519 888 4567 x43484

Fax: 519 725 0160

Email: puremath@uwaterloo.ca

Tuesday, May 5, 2020 — 1:00 PM EDT

**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.

Online

University of Waterloo

200 University Avenue West

Waterloo, Ontario, Canada

N2L 3G1

Departmental office: MC 5304

Phone: 519 888 4567 x43484

Fax: 519 725 0160

Email: puremath@uwaterloo.ca

University of Waterloo

University of Waterloo

43.471468

-80.544205

200 University Avenue West

Waterloo,
ON,
Canada
N2L 3G1

The University of Waterloo acknowledges that much of our work takes place on the traditional territory of the Neutral, Anishinaabeg and Haudenosaunee peoples. Our main campus is situated on the Haldimand Tract, the land granted to the Six Nations that includes six miles on each side of the Grand River. Our active work toward reconciliation takes place across our campuses through research, learning, teaching, and community building, and is centralized within our Office of Indigenous Relations.