Geometric Applications of Linear and Nonlinear Potential Theory

We provide geometric inequalities on $R^n$ and on general manifolds with nonnegative Ricci curvature by employing suitable monotone quantities along the flow of capacitary and $p$-capacitary potentials, as well as through related boundary value problems. Among the main achievements, we cite [(i)] a Willmore-type inequality on manifolds with nonnegative Ricci curvature leading in turn to the sharp Isoperimetric Inequality on $3$-manifolds with nonnegative Ricci curvature ; [(ii)] enhanced Kasue/Croke-Kleiner splitting theorems ; [(iii)] a generalised Minkowski-type inequality in $R^n$ holding with no assumptions on the boundary of the domain considered except for smoothness ; [(iv)] a complete discussion of maximal volume solutions to the least area problem with obstacle on Riemannian manifolds and its relation with the variational $p$-capacity.