We establish several results for the ambient obstruction flow (AOF), a parabolic flow of Riemannian metrics introduced by Bahuaud-Helliwell which is based on the Fefferman-Graham ambient obstruction tensor. The flow may be regarded as a higher order analogue of Ricci flow, and the critical metrics for this flow may be regarded as generalizations of Einstein metrics. First, we obtain local L2 smoothing estimates for the curvature tensor along AOF and use them to prove pointwise smoothing estimates for the curvature tensor. We use the pointwise smoothing estimates to show that the curvature must blow up for a finite time singular solution to AOF. We also use the pointwise smoothing estimates to prove a compactness theorem for a sequence of solutions to AOF with bounded C0 curvature norm and injectivity radius bounded from below at one point. The compactness theorem allows us to obtain a singularity model from a finite time singular solution to AOF and to characterize the behavior at infinity of a nonsingular solution to AOF. Our final result is a rigidity theorem, which states that under suitable conditions a metric that is critical for AOF and has small scale-invariant integral energy has vanishing Riemann curvature tensor.

In this dissertation, we seek to understand prescribed scalar curvature through the gradient flow of conformal metrics. On S^2, we will define a modified Liouville energy and derive a geometric flow equation related to the energy functional. We will prove longtime existence for solutions of this equation with arbitrary data through the methods used by Gursky and Streets. We will then show that Gauss curvature retains its regularity under evolution through the flow assuming bounds on the Gauss curvature. We will finally show that the solution is stable when converging to constant curvature if the initial curvature is close to the geometry of S^2.

This dissertation will show the ways in which generalized geometry elucidates the study of pluriclosed flow. In their 2009 paper, Streets and Tian introduce pluriclosed flow -- a parabolic flow of pluriclosed metrics -- and classify some static solutions. In 2018, Streets expanded this into a geometrization conjecture for compact, complex surfaces. The author is able to use these tools to show an equivalence between pluriclosed flow and a non-linear, coupled Hermitian-Yang-Mills type flow. From there, the author is able to more geometrically prove a result of Streets and Warren -- an Evans-Krylov theorem for pluriclosed flow. The author is also able to use this equivalence to prove long-time existence and convergence of the flow on Bismut-flat manifolds and surfaces of non-negative Kodaira dimension.

We consider various geometric flows which are well adapted for the study of non-K\"ahler complex manifolds. We first study solutions to the pluriclosed flow on compact complex surfaces, giving a complete classification of the long time behavior of homogeneous solutions to the flow and constructing non-trivial, non-K\"ahler expanding soliton solutions to the pluriclosed flow. We also give a simple expression for the evolution of the Lee form on a complex surface, and use this to give simplified proofs of various classification results for fixed points of the flow.

We also consider a functional which is closely related to the Dirichlet energy of maps between two Hermitian manifolds and which has holomorphic maps as global minimizers. We derive it's first and second variations and consider the associated parabolic flow. We provide conditions under which the flow converges to a critical point of the functional and give explicit examples of nice solutions to the flow in Hopf surfaces. We additionally demonstrate so called 'bubbling' criteria for solutions of the flow on surfaces and, using this functional, we give a variational proof that submanifolds of Vaisman manifolds are Vaisman.

In this dissertation, we discuss the behavior of Willmore flow, a fourth-order geometric flow, for complete, properly immersed surfaces in Euclidean space.We develop a-priori estimates for weighted Willmore flows. The estimates are later used to generalize Kuwert and Schäzle’s short-time existence theorem in their 2002 article for complete surfaces with bounded geometry, to find a condition for uniqueness of Willmore flows on complete surfaces, and show gap phenomena of Willmore energy. We also discuss blowups of Willmore flow, as constructed by Kuwert and Schätzle in their 2001 article. We also discuss the Fredholm property of Laplacian operator on the space of normal vector fields, in view of weighted Sobolev spaces as defined by Lockhart in his 1987 article. We give a few conjectures regarding the Fredholm property of linearization of Willmore tensor, Łojasiewicz–Simon inequality, and stability of minimal surfaces with finite energy as Willmore surfaces.