Research Projects

One of the central objects of study in Riemannian geometry is a geometric flow, and a pertinent problem in the study of geometric flows is the classification of its corresponding solitons, which are the self similar solutions to the flow. Perelman (2002) used Hamiton’s Ricci flow with surgery to prove Poincare and Thurston conjectures. This has invoked a great interest in the study of Ricci solitons, which are the singularity models of Hamilton’s Ricci flow. It has also invoked a great amount of interest in using other geometric flows to study problems in Riemannian Geometry. One would like to study flows that have suitable self-similar solutions so comprehending what properties of the flows will induce "nice" properties of solitons is a useful problem to understand. Recently, E. Griffin (2021) introduced the notion of the q-flow (as a generalization of the classical Ricci flow), defined by ∂_t g(t) = q, g(0) = h, where g(t) is a one-parameter family of Riemannian metrics and q is the corresponding symmetric (0, 2) tensor. The associated q-soliton is: \frac{1}{2}L_V g = λg + \frac{1}{2}q, where V is a vector field and λ is a constant. q is essentially a “place-holder” for the tensor which defines the flow that is of interest. Possible examples are the Ricci flow, Bach flow, Cotton flow, Ambient Obstruction flow, Cross Curvature flow, etc. One motivation for defining the q-flow was the previous results of Wylie and Petersen (2022) who generalized a rigidity theorem for homogeneous gradient Ricci solitons to homogeneous gradient solitons to any other isometry invariant curvature flow. Griffin applied these ideas to ambient obstruction solitons and also proved a number of results on solitons associated with the q-flow. Also, Griffin et al. (2025) showed that, any compact ambient obstruction soliton is obstruction flat and has constant scalar curvature. This implies that, on a compact manifold, the ambient obstruction flow has no fixed points up to conformal diffeomorphisms other than the obstruction flat metrics. In the proposed project, we would impose certain restrictions on the associated vector field V of the associated q-soliton and enforce curvature conditions on the manifold, to further probe into its geometry. In dimension 4, the Bach tensor (which has its origins in conformal relativity) and the ambient obstruction tensor are the same, but in higher dimensions the Bach tensor does not have as "nice" (divergence-free and trace-free) general properties as the ambient obstruction tensor. In particular, the past results do not rule out the existence of compact Bach solitons in dimensions 6 and higher. Moreover, we will investigate whether we can obtain general results for q-solitons that will yield results for Bach solitons in higher dimensions. Thus, we expect to demonstrate the versatility of our general findings by applying it to the Bach solitons. Further, we wish to study convergence and stability problems associated with the q-flow.