Date of Award

5-14-2023

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

Advisor(s)

William Wylie

Abstract

Laplacian solitons are self-similar solutions to a geometric flow of $G_2$-structures $\varphi \in \Omega^3(M)$ on smooth $7$-manifolds $M$ called the Laplacian flow. Recently, Laplacian solitons on homogeneous spaces have received increased interest and many new examples have been found by Fernandez-Raffero, Lauret-Nicolini, and others (see, e.g., \cites{FR20, Lau17a, Lau17b, LN20, Nic18}, and \cite{Nic22}). Though there has been recent work on gradient Laplacian solitons in the nonhomogeneous setting due to Haskins and his collaborators (see, e.g., \cites{HN21, HKP22}), very little is known about gradient solitons of a closed Laplacian flow on homogeneous spaces.

In this thesis, we investigate homogeneous closed gradient Laplacian solitons. We prove a Structure Theorem for homogeneous closed gradient Laplacian solitons. We then use the Structure Theorem to ``eliminate'' closed gradient Laplacian solitons. That is, we use the Structure Theorem to show that some closed Laplacian solitons or closed $G_2$-structures cannot be made gradient. We also use the Structure Theorem to obtain the structure of almost abelian solvmanifolds admitting closed gradient Laplacian solitons.

We then study weighted sectional curvature of Riemannian manifolds with density. In particular, we study how weighted sectional curvature bounds give us control over the modified conformal hessian. We use this to prove an inequality resembling the law of cosines, which we call a ``warped law of cosines''.

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