The Bianchi identity and the Ricci curvature equation

Abstract

The main result of this Ph.D. thesis is to present conditions on the curvature and boundary of an orientable, compact manifold under which there is a unique global solution to the Dirichlet boundary value problem (BVP) for the prescribed Ricci curvature equation. This Dirichlet BVP is a determined, non-elliptic system of 2nd-order, quasilinear partial differential equations, which is supplemented with a constraint equation in the form of the so-called Bianchi identity. Indeed, in order to prove the main result of this Ph.D. thesis, we are also required to prove that the kernel of the Bianchi operator is a smooth tame Frechet submanifold of the space of Riemannian metrics on a compact Riemannian manifold with boundary. After presenting an overview of the literature in Chapter 1 and the required notation and background in Riemannian geometry and geometric analysis in Chapter 2, the main results of this thesis are organised into the following two chapters.In Chapter 3 we present conditions under which the kernel of the Bianchi operator is globally a smooth tame Frechet submanifold of the space of Riemannian metrics on a Riemannian compact manifold both with or without boundary. This global submanifold result for the kernel of the Bianchi operator is central to the proof of global existence and uniqueness for the Dirichlet BVP for the Ricci curvature equation, and extends the analogous local submanifold result by Dennis DeTurck in [18] which was fundamental to the proof in [18] of local existence and uniqueness of the prescribed Ricci curvature equation on a compact manifold without boundary.Furthermore, the material in Chapter 3 of this thesis sits more generally within the literature on linearisation stability of nonlinear PDE. Indeed, the method by which we prove that the kernel of the Bianchi operator is a global submanifold of the space of metrics on a compact manifold is an application of the more general technique of proving the linearisation stability of a system of nonlinear PDE, and yields that the Bianchi operator is itself linearisation stable in a sense made precise in Chapter 3.In Chapter 4 we then use the fact that the kernel of the Bianchi operator is globally a smooth tame Frechet manifold, and the Nash-Moser implicit function theorem (IFT) in the smooth tame category, to find and conditions under which globally there is a unique Riemannian metric satisfying the Dirichlet BVP for the Ricci curvature equation. This global existence and uniqueness result is motivated by, and can be viewed as a modification of, an analogous result for Einstein manifolds of negative sectional curvature and convex and umbilical boundary presented by Jean-Marc Schlenker in [50]. Loosely speaking, in the context of Einstein manifolds the kernel of the Bianchi operator is the entire space of Riemannian metrics and thus automatically a smooth tame Frechet manifold; however, in the context of the prescribed Ricci curvature equation, in which this Ph.D. thesis is interested, we are required to prove that the kernel is a smooth tame Frechet submanifold in order to apply the Nash-Moser IFT.In general, global existence for the prescribed Ricci curvature equation on a manifold with or without boundary is difficult and results in this direction are few; the main goal of this thesis makes an important contribution in this area and the techniques used to prove it have potential applications in other areas of geometric analysis such as Yang-Mills gauge theory and curvature flows.