We investigate compactness phenomena involving free boundary minimal hypersurfaces in Riemannian manifolds of dimension less than eight. We provide natural geometric conditions that ensure strong one-sheeted graphical subsequential convergence, discuss the limit behaviour when multi-sheeted convergence happens and derive various consequences in terms of finiteness and topological control.Mathematics Subject Classification (MSC 2010): Primary 53A10; Secondary 53C42, 49Q05.

We perform an optimal localization of asymptotically flat initial data sets and construct data that have positive ADM mass but are exactly trivial outside a cone of arbitrarily small aperture. The gluing scheme that we develop allows to produce a new class of N -body solutions for the Einstein equation, which patently exhibit the phenomenon of gravitational shielding: for any large T we can engineer solutions where any two massive bodies do not interact at all for any time t ∈ (0, T ), in striking contrast with the Newtonian gravity scenario.

The study of stable minimal surfaces in Riemannian 3-manifolds (M, g) with non-negative scalar curvature has a rich history. In this paper, we prove rigidity of such surfaces when (M, g) is asymptotically flat and has horizon boundary. As a consequence, we obtain an effective version of the positive mass theorem in terms of isoperimetric or, more generally, closed volume-preserving stable CMC surfaces that is appealing from both a physical and a purely geometric point of view. We also include a proof of the following conjecture of Schoen: An asymptotically flat Riemannian 3-manifold with non-negative scalar curvature that contains an unbounded area-minimizing surface is isometric to flat R 3 .

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Given a closed two dimensional manifold, we prove a general existence result for a class of elliptic PDEs with exponential nonlinearities and negative Dirac deltas on the right-hand side, extending a theory recently obtained for the regular case. This is done by global methods: since the associated Euler functional is in general unbounded from below, we need to define a new model space, generalizing the so-called space of formal barycenters and characterizing (up to homotopy equivalence) its very low sublevels. As a result, the analytic problem is reduced to a topological one concerning the contractibility of this model space. To this aim, we prove a new functional inequality in the spirit of [16] and then we employ a min-max scheme based on a cone-style construction, jointly with the blow-up analysis given in [5] (after [6] and [8]). This study is motivated by abelian Chern-Simons theory in self-dual regime, or from the problem of prescribing the Gaussian curvature in presence of conical singularities (hence generalizing a problem raised by Kazdan and Warner in [26]).with ρ a real parameter and h : Σ → R a smooth function. However, one basic feature of this geometric problem is that such a ρ = K g is related to the topology of Σ by means of the Gauss-Bonnet formula Σ K g dV g = 2πχ(Σ).Once we assume, without loss of generality, that V ol g (Σ) = 1, we have that this equation forces K g to attain values that are (some) integer multiples of 4π: therefore, on Riemann surfaces, we say that K g is 1