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We begin by studying a property first introduced by Roger Penrose as a consistency condition for possible constructions of quantum gravity. In an asymptotically flat spacetime, this property, which we call the Penrose property, is satisfied if for any two endless timelike curves λ, ν in the same domain of outer dependence, there exists a future directed timelike curve from λ to ν. Penrose showed that this property is equivalent to the entirety of future null infinity being contained in the timelike future of every point in past null infinity. We will investigate how the Penrose property is affected by the mass and dimensionality of spacetime. By considering various examples, we find that this property appears to be a property of positive mass spacetimes in 3 and 4 dimensions. We will then move on to study the Penrose property in greater generality. In particular we will consider how this property can be generalised to spacetimes with a non-zero cosmological constant. We find that in asymptotically de Sitter spacetimes the property can remain essentially unchanged, however in asymptotically anti-de Sitter spacetimes it is necessary for it to be re-stated in a way which is more suited to spacetimes with a timelike boundary. In the latter case we arrive at a property previously considered by Gao and Wald. Curiously, this property was shown to fail in spacetimes which focus null geodesics. This is in contrast to our findings in asymptotically flat and asymptotically de Sitter spacetimes. Next we show how some of the methods established in the study of the Penrose property can be used to prove a version of the positive mass theorem in higher dimensions. This proof is inspired by an argument of Penrose, Sorkin and Woolgar in 3+1 dimensions. This argument relied on causality arguments to show that spacetimes which focus null geodesics (a feature we expect to be indicative of positive mass spacetimes) cannot have negative ADM mass. Penrose has argued that the Penrose property is a property of spacetime near i⁰. We have also demonstrated how this property appears to depend on the dimensionality of spacetime. Motivated by this, we study more generally the nature of possible conformal completions at spatial infinity. In particular, we show that (d + 1) dimensional Myers-Perry metrics (d ≥ 4) have a conformal completion at spatial infinity of C^{d−3,1} differentiability class, and that this result is optimal in even spacetime dimension in the sense that no C^{d−2} completion exists unless the ADM mass is zero. We also present the associated asymptotic symmetries. Finally, we will study conformal geodesics. We will show that these curves cannot spiral towards a point. This resolves a major unsolved problem in the field. We will begin by developing a proof of the same result for metric geodesics which does not rely on length minimisation arguments, before showing how this proof can be generalised to rule out spiralling of conformal geodesics.

In this thesis we obtain new estimates and regularity results for some fully nonlinear elliptic equations arising in conformal geometry. Our first set of new results concern local pointwise second derivative estimates for elliptic solutions to the σk-Yamabe equation. In Chapter 2 we obtain such estimates for W2,p-strong solutions on Euclidean domains, addressing both the so-called positive and negative cases. We explore two methods for obtaining these estimates: an integrability improvement argument coupled with Moser iteration, and a method using the Alexandrov-Bakelman-Pucci estimate. Our estimates are obtained in the more general context of augmented Hessian equations. In Chapter 3 we obtain similar estimates for smooth solutions on manifolds when k = 2. Our work here contributes to a growing literature on the regularity theory for the σk-Yamabe equation and, from a broader perspective, the regularity theory for fully nonlinear, non-uniformly elliptic equations. In Chapter 4 we study the existence of conformal metrics satisfying g-1 Agτ ∈ Γ2+, where Agτ is the trace-modified Schouten tensor. When τ = 1, this is an important question in the context of the σ2-Yamabe problem, and it is also of independent geometric and topological interest when τ ≤ 1. Our focus will be on three dimensions; we prove a new existence result when τ < 1, and obtain a new proof of a result of Ge, Lin & Wang [GLW10] when τ = 1, with an eye towards tackling some related problems. In Chapter 5 we obtain integral estimates for a fourth order perturbation of the (trace-modified) σ2-Yamabe equation in three dimensions, in the spirit of Chang, Gursky & Yang [CGY02b]. Our study of this equation is partly motivated by the existence problems considered in Chapter 4, but also from the analytic viewpoint of using fourth order regularisations to study non-uniformly elliptic PDEs of second order.

The Riemannian submersion π : SO_0(1,n) ->H^n is a principal bundle and its fiber at π (e) is the imbedding of SO(n) into SO_0(1,n) , where e is the identity of both SO_0(1,n) and SO(n). In this study, we associate a curve, starting from the identity, in SO(n) to a given piecewise smooth surface with boundary, homeomorphic to the closed disk D^2, in H^n such that the starting point and the ending point of the curve agree with those of the horizontal lifting of the boundary curve of the given surface with boundary, respectively, and that the length of the curve is as same as the area of the given surface with boundary. In addition, the curve in SO(n) relates the connection of its tangent vector to the curvature of some point in SO_0(1,n).