This thesis proves rigidity theorems for three-dimensional Riemannian manifolds with scalar curvature bounded below by 6K, where K ϵ {-1,0,1}, by placing restrictions on the Hawking mass of surfaces. The primary tool will be the geometry of perturbed geodesic spheres.

The majority of the work focusses on the case K = 0, for which we prove both a local and global rigidity result. The first states that if every point in an open subset Ω has a neighbourhood U ϲ Ω such that the supremum of the Hawking mass of surfaces contained in U is non-positive, then Ω is locally isometric to Euclidean R3. Taking W to be the ambient manifold and further assuming it is asymptotically locally simply connected (which encompasses as a special case, the standard asymptotically flat property), we will prove that it must be globally isometric to Euclidean R3. The method involves computing the Taylor expansion of the Hawking mass of optimal perturbed geodesic spheres, which will be positive when the space is non-flat. This will allow us to prove a positive lower bound (comprised of curvature tensors) on the Bartnik mass of (non-flat) open sets, once we prove that perturbed geodesic spheres are outer-minimising.

The proof of the outer-minimising property requires the framework of sets of finite perimeter. Specifically, for each element of a sequence of shrinking perturbed geodesic spheres, we consider the corresponding set with least perimeter that contains it. We prove convergence and regularity properties for this new sequence and determine that, eventually, the boundaries of its elements are the original spheres.

Later, we will extend some of our results to the case K≠0, where the model space is the complete, simply connected Riemannian manifold of constant sectional curvature K. In order to extend the global rigidity
theorem to the case K = -1, we consider an alternative asymptotic condition, namely the global asymptotic volume property, which compares the volume of large balls to those in the model space.