Abstract:
The aim of this seminar is to describe, through a couple of examples coming form classical potential theory and general relativity, the interplay between some basic Riemannian splitting principles and the rigidity case of relevant geometric inequalities, such as the Willmore-Chern inequality for compact hypersurfaces in Euclidean space or the Riemannian Penrose inequality for static black holes. As a counterpart of this fact, we will see that, beside the characterization of the rigidity case, our methods can be used to produce new monotonicity formulas, which imply in turn the inequality under consideration in its full generality.
The results presented are obtained in collaboration with V. Agostiniani.