RESEARCH TOPICS:

I work in the field of geometric analysis, more precisely my main focus is the study of PDEs using geometric methods. The two main topics I am currently interested in are the following:

• Static metrics in general relativity: The main topic of my PhD thesis has been the study of static metrics in General Relativity. These are solutions of the Einstein Field Equations in the vacuum admitting a global irrotational Killing vector field. Under this assumption, one is led to study a PDE problem on a Riemannian manifold depending on a parameter Λ called cosmological constant. The analysis of this problem differs significantly depending on whether Λ is positive, negative or null, also in view of the fact that the physically relevant boundary/asymptotic conditions are different in the three cases. 

In the case Λ = 0, a common assumption is that the manifold is asymptotically flat. Under this hypothesis, and supposing that the boundary is nonempty, it is known that the solution is necessarily isometric to the Schwarzschild solution. This is the content of the classical Black Hole Uniqueness Theorem, proven by Israel, Robinson, Bunting and Masood-ul-Alam, using the Positive Mass Theorem of Schoen and Yau, Witten for the ADM mass as a fundamental tool.

Much less is known in the case of a positive cosmological constant. For this reason, in a series of three works, me and my thesis advisor L. Mazzieri focused our study on the case Λ > 0. In particular, we introduced a notion of mass, and we proved that it satisfies a Positive Mass Statement and a Riemannian Penrose–like inequality. Building on this, we were also able to prove a uniqueness result for the Schwarzschild–de Sitter solution.

• Overdetermined boundary value problems: It is well known that a large class of PDE problems on the inner or the outer region of a bounded domain in Euclidean space admits a unique solution. For such PDE problems, if one requires some additional condition to be satisfied by the domain or by the target function, then the problem will continue to admit a solution only for special domains. The study of such overdetermining conditions is an interesting and well researched topic, as it allows to prove inequalities satisfied by the solutions and, even most notably, sometimes leads to the proof of completely geometric inequalities for domains in the Euclidean space.

In a paper with G. Mascellani and L. Mazzieri, we studied the potential problem of a charged body, that is, the analysis of harmonic functions outside a bounded domain. In our work we have found new overdetermining conditions forcing the rotational symmetry of the solution, as well as alternative proofs of some known results.

More recently, with L. Mazzieri and V. Agostiniani, we are studied the torsion problem, that is, the analysis of function with constant (nonzero) laplacian inside a bounded domain. With an analysis strikingly similar to the one employed to study static metrics, we are able to derive some new overdetermining conditions, as well as some purely geometric inequalities on the domain.

FUTURE DEVELOPMENTS: 

• From static to stationary: A natural project is that of adapting our study of static metrics to the more general stationary case, that is, the case of spacetimes admitting a timelike Killing vector field which is not necessarily irrotational. In this framework, a much richer family of explicit solutions is known (the Kerr–de Sitter spacetimes). We expect that it should be possible to define a notion of mass in this framework, which is coherent with the one we introduced in the static case. The next step would be to study this mass and hopefully prove that it still satisfies a Positive Mass Statement and a Riemannian Penrose–like inequality, with the final aim of deriving a uniqueness result for the Kerr–de Sitter spacetimes.

• Min-Oo's conjecture: Some good notions of mass for asymptotically flat and asymptotically hyperbolic manifolds are already known, and they satisfy a Positive Mass Theorem, meaning that, under the assumption of a natural lower bound for the scalar curvature, the mass is always positive, and it is zero if and only if the manifold is isometric to the flat (or the hyperbolic) space form. A well known conjecture by Min-Oo, who proposed a characterization of the round metric among metrics on the hemisphere with curvature bounded from below by a positive constant, has been regarded for years as the spherical analogue of the rigidity statements of the above mentioned Positive Mass Theorems. Min-Oo's conjecture was thought to be true for a long time until it was finally disproven by Brendle, Marques and Neves in every dimension > 2. This negative result suggests that there is no straightforward ways of transfering in the spherical framework the notions of mass developed in the asymptotically flat and hyperbolic setting. On the other hand, our Positive Mass Statement can be regarded as a characterization of the round hemisphere among static metrics, in a spirit similar to Min-Oo's. It would be interesting to understand if our approach can be generalized (for instance to the already mentioned stationary case, or more ambitiously to metrics with scalar curvature bounded from below by a positive constant) to prove an effective version of Min-Oo's conjecture.

• Static metrics with negative cosmological constant: It seems natural to try to employ the method used in our analysis of the positive cosmological constant case in order to study static metrics with negative cosmological constant. The ultimate purpose would be that of proving a characterization of the Schwarzschild–Anti de Sitter spacetime.

• Pólya-Szegő conjecture: A well known conjecture by Pólya and Szegő states that the capacity of a bounded convex set with positive surface measure is greater than or equal to some function of the surface measure of the domain. Moreover, the conjecture states that the equality holds if and only if the domain is a round 2-dimensional disk. This problem is one of our main motivation for continuing our study of the potential problem of a domain, which has already been useful to provide some inequalities involving the capacity. It should be noticed that, up to now, in our study we were interested in finding overdetermining conditions under which the domain is forced to be rotationally symmetric. In order to prove the Pólya-Szegő conjecture, we expect that we should modify our method in order to detect disks instead of balls.

• Torsion problem for unbounded domains: From some preliminary computations, it appears that, when the domain is not compact, the torsion problem should be the natural Euclidean analogue of the problem of static metrics with negative cosmological constant. Therefore, before venturing into the study of static metrics with negative cosmological constant, as a first step it seems reasonable to focus on the study of the torsion problem for unbounded domains, which should be an easier task.