The aim of this project is two-fold. The volume preserving mean curvature flow shrinks the area of a surface as fast as possible but keeps the enclosed volume fixed. Based on previous work and recent development in the field we would like to understand the Neumann free boundary setting inside of a convex domain for curves. We are particularly interested whether it is possible to exclude singularities. Furthermore, we would like to study the connection of limits of the flow without singularities to the solution of the relative isoperimetric problem.
The second part of this project is connected to the Willmore functional. For example, it is interesting to know how many ways there are to deform a Willmore surface in a way that decreases the Willmore energy. This is a Morse theoretical questions. Recently, the Morse Index of Willmore spheres that arise as inverted complete minimal surface with embedded planar ends was computed. Amongst other things, several asymptotic geometric properties of the minimal surface at infinity have consequences for Morse theoretical questions. We would like to deepen the study of these connections. Furthermore, we are interested in related questions to the Sphere Eversion. In particular, we want to understand asymptotic stability of non-compact limit surfaces for singularities of the Willmore flow.