Monotonicity Formulas in Nonlinear Potential Theory and their geometric applications

Venue: The event will take in presence and online through the ZOOM platform. To get the access codes please contact the secretary office (phd.maths [at] unitn.it)
Time: 10:00 am

Luca Benatti - PhD Student in Mathematics

Abstract: The Minkowski Inequality is a classic geometric inequality that provides a lower bound of the total mean curvature in terms of the area of a convex subset of the flat Euclidean space, with equality holding on spheres. In the last century, many authors wondered if this geometric inequality was possibly extended both to a larger class of subsets and ambient manifolds. In this final talk, we present the derivation of our sharp Minkowski Inequality in the setting of Riemannian manifolds with nonnegative Ricci curvature as a consequence of the Monotonicity Formulas holding along the flow of the level sets of the p-capacitary potential. Along with the proof we also analise the deep connections between the Nonlinear Potential Theory and the weak formulation of the Inverse Mean Curvature Flow, which is at the base of the proof of the rigidity in the Minkowski Inequality. We conclude proposing some further reasearch lines scaturating from this study.

Supervisor: Lorenzo Mazzieri (University of Trento)