ETD

• geometric inequalities
• imcf
• monotonicity formulas
• p-potential theory

It is well known that the study of geometric flows, such as the inverse mean curvature flow (IMCF), plays a central role in deriving geometric inequalities. A classical approach, exemplified by Huisken and Illmanen in their proof of the Riemannian Penrose inequality, is to work directly with the IMCF. This, however, is technically demanding, as the flow must be weak enough to exist in the relevant setting. An alternative perspective was introduced by Moser, who established a link between IMCF and p-harmonic potentials. Since then, computations performed in the p-harmonic framework and subsequently translated to the IMCF have become a powerful tool in geometric analysis. In this thesis we adopt this approach to study a family of geometric inequalities: we recall the main properties of p-harmonic potentials and the IMCF, discuss their connection, carry out the necessary computations in the p-harmonic setting, and show how they translate to the flow, concluding with examples that illustrate the effectiveness of this method.