Old and new Sobolev inequalities on manifolds via the ABP method

In the first part of the talk, we will survey the implementation of the ABP method to prove (sharp) isoperimetric inequalities both in Euclidean spaces and on complete Riemannian manifolds with nonnegative curvature and Euclidean volume growth, as well as on minimal submanifolds therein. In particular we will outline the breakthrough contributions by X. Cabré and S. Brendle.
In the second part, we will present a recent observation regarding the method, which allows us to recover classical Sobolev and Michael-Simons inequalities on manifolds, as well as to obtain new ones.