Geometric flows and special holonomy

Special holonomy is a central topic within Geometry, since it currently provides the only non-trivial mechanism for constructing compact spaces which are Ricci-flat: such spaces go back to the very beginnings of the study of curved spaces (i.e. Riemannian geometry) and provide analogues of solutions to Einstein's vacuum gravity field equations from General Relativity.

The study of special holonomy has clear connections to numerous areas, including Algebra, Analysis, Topology and Mathematical Physics, so work in this area has the potential to have wide impact to a diverse range of researchers.

Finding special holonomy spaces which are not complex (so-called exceptional holonomy spaces) is very challenging. For each of these exceptional holonomies, G2 in dimension 7 and Spin(7) in dimension 8, there is only one known method, due to Dominic Joyce, which is an elliptic "small perturbation" technique. Though this method is very powerful, it is quite limited as one finding initial geometries to apply the technique is difficult as they have to be "close to degenerate".

One might hope instead to use geometric flows to study exceptional holonomy, since geometric flows have proven due to be powerful tools in Riemannian geometry, including in applications to other areas such as Topology. There has been a proposal for an alternative approach to study the existence problem for exceptional holonomy in the G2 setting due to Robert Bryant, called the G2-Laplacian flow, which has and continues to receive significant attention.

However, in the Spin(7) case, there is no analogue to Bryant's flow and relatively little has been done in this setting in comparison to the G2 case. Recently, a Spin(7)-flow has been introduced and studied, which is the gradient flow for the L2-norm of the torsion, but very little is known about this flow.

The focus of the project is to study this Spin(7) torsion flow. The project will begin by considering the homogeneous setting, looking at explicit examples, searching for solitons and studying long-time behaviour and singularity formation. The next aim will be to look at various dimensional reductions of the flow, potentially including 1, 4, 6 or 7 dimensions.

The first two cases (1 and 4 dimensions) provide areas where the strongest results can potentially be obtained, such as non-trivial long-time existence results. The latter two cases (6 and 7 dimensions) should have interesting connections to complex and G2 geometry. The final aim will be to make progress on flows in the 6 and 7-dimensional setting, either related to the Spin(7) flow, or the G2-Laplacian flow.

The flow is only recently introduced and has not been studied using homogeneous methods or dimension reduction thus far. The work in this project therefore is entirely novel both in its conception and in its methodology. The link to flows in dimensions 4, 6 and 7 is particularly interesting and novel.

This project falls within the EPSRC Geometry and Topology research area within Mathematical Sciences.

No companies are involved in the project. No collaborators are currently involved in the project, though it is possible that collaborators may join the project at a later stage depending on the precise outcomes and progress in the project.