Metrics of special curvature in differential geometry

This project is in the area of differential geometry, the branch of mathematics primarily concerned with the length and curvature properties of smooth objects (called manifolds), often in higher dimensions. Differential geometry is a key part of modern mathematics with direct and close links to numerous areas including algebra, analysis, topology and theoretical physics.

A central area of study in differential geometry is Riemannian manifolds: smooth objects endowed with a Riemannian metric, which allows one to measure lengths and angles. The curvature properties of a Riemannian manifold are encoded in the Riemann curvature tensor. This depends in a complicated and nonlinear way on the metric, and so curvature conditions involve nonlinear partial differential equations which are usually difficult to analyse. Examples include the Einstein equations and the Ricci soliton equations, both of which are key equations in differential geometry because of their potential links to topology and mathematical physics, as well as to geometric analysis through the Ricci flow. Other important equations on Riemannian manifolds also involve auxiliary data such as connexions or sections of vector or spinor bundles on the manifold.

This project will involve the use of symmetry assumptions to simplify such curvature equations by reducing them to systems of ordinary differential equations or partial differential equations of lower order. Equations to be considered include the Einstein equation and also the Strominger system on a Hermitian or Kähler manifold. The Einstein equation is of fundamental importance in Riemannian geometry, yet our understanding of solutions is relatively poor in dimensions above four, and this is due in poor to a lack of examples. The Strominger system originated in mathematical physics through the study of superstrings in torsion backgrounds, and involves a Hermitian metric and a connexion on the manifold. Although it is well-motivated and studied both by mathematicians and physicists, the paucity of non-trivial examples of solutions to the Strominger system again means that there is a clear gap in our understanding of the system.

The methodology will involve using the theory of homogeneous spaces and Lie groups to implement the symmetry reduction, as well as using techniques from nonlinear dynamical systems and elliptic partial differential equations to analyse the resulting systems. This methodology, in particular, has not been used to study the Strominger system, and so represents a novel research direction. The methodology has proved to be very powerful in other related situations, and so one can hope that it will be similarly fruitful to utilize this tool in the setting of the project.

The project should lead to a better understanding of these equations, including new existence results for solutions, possible explicit solutions in some cases, and results on uniqueness and moduli. The results will be of relevance not only to pure mathematicians working in differential geometry but also to mathematical physicists.

This project falls within EPSRC research area Geometry and Topology, but also has strong links to Analysis and Mathematical Physics.