Geometric Techniques in PDE Theory, Fluid Dynamics, and Relativity

In this thesis, we present a number of recently published results [1–3] in the field of geometry, motivated by the study of partial differential equations arising in fluid dynamics and relativity. In this format, shared background material is consolidated and the results are supplemented by additional exposition. This thesis is divided into two parts. In the first, we introduce a general framework for studying incompressible Navier–Stokes flows on a Riemannian manifold via Monge–Ampère geometry, inspired by multi-symplectic techniques. In two dimensions, the Poisson equation for the pressure, which acts as a diagnostic equation for the balance of vorticity and rate-of-strain, can be encoded in terms of a Monge–Ampère structure on the cotangent bundle and we show that this structure generalises to a pair of multi-symplectic forms in higher dimensions. Submanifolds of the cotangent bundle on which these forms vanish are subsequently studied in lieu of solutions and are equipped with a metric whose signature and curvature are dictated by the accumulation of vorticity. This description admits a (multi-)symplectic reduction principle for three dimensional flows with symmetry and allows topological information about the flow to be deduced. We conclude by discussing ongoing work to classify the higher dimensional generalisation of a Monge–Ampère structure. In the second part of this thesis, we derive analogues of Alexandrov’s patchwork theorem and Toponogov’s theorem for globalising curvature bounds defined by comparison methods, in the rapidly developing field of Lorentzian pre-length spaces. Along the way, we derive a number of supplementary results, including Lorentzian analogues of the Bonnet–Myers theorem and the Lebesgue number lemma. We also highlight several small, but not insignificant, modifications to the definition of a comparison neighbourhood to account for points with infinite time separation (e.g. in Anti-de Sitter space). We conclude by discussing the open problem of the relationship between Lorentzian pre-length spaces, causal sets, and Gromov–Hausdorff convergence.