Quermassintegral preserving local curvature flows

• Applicant: Professor Dr. Julian Scheuer
• Subject Area: Mathematics
• Term: from 2018 to 2020
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 400729345

We consider inverse curvature flows with forcing terms of closed graphical and mean-convex hypersurfaces in a class of cylindrical warped ambient manifolds, which contains the simply connected spaceforms. For example, by adding suitable combinations of the radial distance and angle functions to the inverse mean curvature flow equation, one obtains a surface area preserving flow which decreases the total mean curvature. The special feature of this flow is that, contrary to previous quermassintegral preserving flows, it is local and does not contain a nonlocal forcing term. This feature has several technical benefits. The aim of this project is to deduce the long-time existence of these flows and smooth convergence to a coordinate slice for graphical and mean-convex initial hypersurfaces. As applications several generalisations of classical Alexandrov-Fenchel inequalities would follow for non-convex hypersurfaces. A Heintze-Karcher type inequality for closed mean-convex hypersurfaces has proven to be a helpful tool in the deduction of monotone quantities along such flows. This inequality holds with precise equality if and only if the hypersurface is a coordinate slice. A stability version of this result would be very useful in the investigation of the asymptotics of such curvature flows and hence shall also be deduced within this project.

• DFG Programme: Research Fellowships
• International Connection: USA
• Participating Institution: Columbia University in the City of New York ; Department of Mathematics
• Host: Professor Dr. Simon Brendle