In this note we survey and compare the monotonicity formulas recently discovered by the authors in  and  in the context of classical potential theory and in the study of static metrics, respectively. In both cases we discuss the most significant implications of the monotonicity formulas in terms of sharp analytic and geometric inequalities. In particular, we derive the classical Willmore inequality for smooth compact hypersurfaces embedded in Euclidean space and the Riemannian Penrose inequality for static Black Holes with connected horizon.
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Virginia Agostiniani, Lorenzo Mazzieri, Comparing monotonicity formulas for electrostatic potentials and static metrics. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 28 (2017), no. 1, pp. 7–20DOI 10.4171/RLM/749