Geometrie und Analysis unter verallgemeinerten Krümmungsbegriffen

01/09/2017 to 31/08/2019

The main purpose of this proposal is twofold. In the first part we try to go beyond the current framework of weighted Ricci curvature on manifold: Besides trying to extend the non-smooth techniques in the presence of boundaries, the project tries to understand the Bakry-\'Emery Ricci tensor for $1$-forms and for non-conventional dimensions. The second part tries to understand isopermetry: Using the behavior at infinity in form of asymptotic flatness and local behavior in terms of scalar curvature, we hope to get strong local-to-global implications on the asymptotic volume growth. Another approach treats isoperimetric comparison principles in a purely synthetic way. Adding necessary conditions that allow a first order analysis, we hope to understand geometric and analytic implications that follow directly from the isoperimetric conditions.