The aim of the project is to shed light on some relevant geometric aspects arising in potential theory. In concrete, we are interested in studying geometric properties of a domain in connection with the related capacitary or p-capacitary potential, both in the Euclidean and in a more general Riemannian context.
The archetypical case is the one analysed in , where some new monotonicity formulas associated with the level set flow of the electrostatic potential in the Euclidean space are deduced. Such formulas are then applied to deduce a new quantitative estimate of the Willmore Inequality.
This point of view and the developed technique have already been fruitfully employed to various different contexts: to that of static metrics in general relativity [3,4,5], to the case of manifolds with nonnegative Ricci curvature , to the case of the Euclidean p-capacitary potential .
Here are, in a nutshell, the problems which we would like to consider in our project: Minkowski inequality for mean convex domains; Riemannian Penrose Inequality in low dimension; Monotonicity formulas in Cartan-Hadamard manifolds; Torsion problem with boundary made of more than one connected components; An alternative approach to the Willmore Conjecture.
 V. Agostiniani, M. Fogagnolo, and L. Mazzieri. Sharp geometric inequalities for closed hypersurfaces in manifolds with nonnegative ricci curvature. arXiv:1812.05022.
 V. Agostiniani and L.Mazzieri. Monotonicity formulas in potential theory. arXiv:1606.02489v3.
 V. Agostiniani and L. Mazzieri. On the geometry of the level sets of bounded static potentials. Commun. Math. Phys., 355: 261-301, 2017.
 S. Borghini and L. Mazzieri. On the mass of static metrics with positive cosmological constant: II. arXiv:1711.07024.
 S. Borghini and L. Mazzieri. On the mass of static metrics with positive cosmological constant: I. Classical Quantum Gravity, 35(12): 125001, 43, 2018.
 M. Fogagnolo, L. Mazzieri, and A. Pinamonti. Geometric aspects of p-capacitary potentials. To appear on Ann. Inst. H. Poincaré Anal. Non Linéaire. https://doi.org/10.1016/j.anihpc.2018.11.005.