Una dimostrazione della disuguaglianza Riemanniana di Penrose mediante flusso conforme

The purpose of the thesis is to investigate a proof of the Riemannian Penrose inequality in dimension less than eight, due to H. L. Bray and D. A. Lee, which exploits a conformal flow of metrics. The Riemannian Penrose inequality is a particular case of a more general statement: the Penrose conjecture. Penrose came up with the conjecture named after him in the sixties while working on the more ambitious idea of cosmic censorship, according to which there are no so-called "naked singularities" (in gravitational collapse singularities should always be "protected" by the event horizon of the black hole). Loosely speaking, in the Hamiltonian approach to General Relativity a notion of mass of a given (asymptotically flat) spacelike slice of spacetime, called the ADM mass, was highlighted. The general Penrose inequality is a statement which provides a lower bound for the ADM mass in terms of the area of the black holes in the slice itself. When we consider a stronger assumption on the foliation of spacetime, namely "time symmetry" of the initial data set, the natural physical assumption that the energy-momentum vector be causal (i.e. the dominant energy condition) becomes a constraint on the scalar curvature of the slice of spacetime. For this reason in the literature this particular case is known as "Riemannian" Penrose inequality. The Riemannian Penrose inequality remains conjectural in dimension bigger or equal than eight. The thesis is organized as follows: first of all we give the basic definitions in order to set the problem and understand the statement, then we give the idea of the conformal flow of metrics and why this kind of flow is a useful tool for the proof. Then we prove that the flow of metrics we built exists and converges to a Schwarzschild metric; after that, except for various technical details left, our main result follows.