This project wants to study the behavior of geometric objects in situations that include limiting or degenerating properties.
The first object are immersed closed surfaces that degenerate in moduli space. For oriented surfaces it is known that the Willmore energy cannot drop under a certain level. This geometric property of surfaces at the boundary of moduli space was used to show compactness of a certain sequence of immersions if one can make sure to stay below that energy level. A similar result was obtained for Klein bottles. The project wants to study non-orientable closed surfaces degenerating in moduli space also for higher (non-orientable) genera.
The second object are surfaces or curves that move under the volume preserving mean curvature flow. Among the objectives of this project are a criterion that guarantees the appearance of a singularity for the flow with Neumann free boundary conditions, the comparison of the properties of these singularities to other geometric flows and other/no boundary conditions, and the study of limits of the flow without singularities, in particular with respect to the relative isoperimetry problem.
We give a topological classification of Lawson's bipolar minimal surfaces corresponding to his ξ- and η-family. Therefrom we deduce upper as well as lower bounds on the area of these surfaces, and find that they are not embedded.
We obtain estimates on nonlocal quantities appearing in the Volume Preserving Mean Curvature Flow (VPMCF) in the closed, Euclidean setting. As a result we demonstrate that blowups of finite time singularities of VPMCF are ancient solutions to Mean Curvature Flow (MCF), prove that monotonicity methods may always be applied at finite times and obtain information on the asymptotics of the flow.
We demonstrate that the property of being Alexandrov immersed is preserved along mean curvature flow. Furthermore, we demonstrate that mean curvature flow techniques for mean convex embedded flows such as noncollapsing and gradient estimates also hold in this setting. We also indicate the necessary modifications to the work of Brendle-Huisken to allow for mean curvature flow with surgery for the Alexandrov immersed, 2-dimensional setting.