Quermassintegral preserving local curvature flows
Final Report Abstract
This project dealt with hypersurface flows the speed of which is composed from the curvature and lower order quantities such as the support and distance function of the hypersurface. The main speeds of interest are designed to keep a certain geometric quantity (a certain quermassintegral) fixed, while decreasing another one. The aim of the project was to analyse these flows and possibly prove good asymptotic behaviour, namely convergence to a geodesic sphere. In particular in non-Euclidean ambient spaces, this would then yield new and important geometric inequalities, so-called Alexandrov-Fenchel inequalities. The precise goals of this project were achieved partially. The full set of desired Alexandrov-Fenchel inequalities, in particular in the hyperbolic space and the sphere, could not be proved. However, specific cases could be verified to an extent that exceeded expectations. In particular, a new Minkowski inequality was obtained in a class of warped products that are not asymptotically of constant curvature. Such a result is surprising because previous techniques required precisely this property in order to make flow approaches work. In addition to that we obtained first results of this kind in Lorentzian ambient spaces, which prospectively might be of interest in general relativity. Namely we obtained a new Minkowski inequality in the Lorentzian de Sitter space, as well as new isoperimetric inequalities in generalized Robertson-Walker spaces. Another problem investigated in the project was a stability version of the well-known Heintze-Karcher inequality in spaceforms. It asks for an error estimate for the closeness of a hypersurface to a sphere in dependence of the deficit in the Heintze-Karcher inequality. As the techniques developed for the proof of this result yield a much broader range of applications, such as stability in non-convex Alexandrov-Fenchel inequalities, the work on this question is ongoing in a late stage.
Publications
- The Minkowski inequality in de Sitter space (2019)
Scheuer, Julian
- Isoperimetric problems for spacelike domains in generalized Robertson-Walker spaces, J. Evol. Equ., (2020)
Lambert, Ben and Scheuer, Julian
(See online at https://doi.org/10.1007/s00028-020-00584-z) - Minkowski inequalities and constrained inverse curvature flows in warped spaces, (2020), Adv. Calc. Var.
Scheuer, Julian
(See online at https://doi.org/10.1515/acv-2020-0050) - Orlicz-Minkowski flows, Calc. Var. Partial Differ. Equ., (2020)
Bryan, Paul; Ivaki, Mohammad and Scheuer, Julian
(See online at https://doi.org/10.1007/s00526-020-01886-3)
