Asymptotical behavior of volume preserving mean curvature flow and stationary sets of forced mean curvature flow

Abstract

The main subject of this dissertation is mean curvature type of flows, in particular the volume preserving mean curvature flow. A classical flow in this context is seen as a smooth time evolution of n-dimensional sets. An important question is when a given mean curvature type of flow exists at all times, and thus does not form singularities. A singularity of a flow is a time where one cannot continue the flow, and usually the evolving set experiences topological changes. The work consists of three articles. In the first article [A], the focus lies on a behavior of a volume preserving mean curvature flow starting nearby a so-called strictly stable set in a three- or four-dimensional flat torus. The contribution of the first article is to show that if the previous flow starts sufficiently close to the strictly stable set in the H3-sense, then the flow exists at all times and converges, up to a small translation, to the set at an exponential rate. In particular, such a flow does not experience singularities. The second article [B] and the third article [C] concern generalizations of mean curvature type of flows, so-called flat flows, obtained via the minimizing movement method. Advantages of such a generalization are that it is defined at all times and requires less regularity for a given initial set compared to a mean curvature type of flow. In [B], it is shown that a flat flow of volume preserving mean curvature flow, starting from a bounded set of finite perimeter, has a shape of a finite union of equisized balls with mutually disjoint interiors in the asymptotical sense. The previous result relies on a new quantitative Alexandrov’s theorem, also proven in [B]. This theorem says that if a bounded C2-regular set, with a fixed upper bound on perimeter and a fixed lower bound on volume in an n-dimensional Euclidean space, has a boundary mean curvature close to a constant value in the Ln−1-sense, then the set is close to a finite union of equisized balls, with mutually disjoint interiors, in the Hausdorff-sense. In [C], it is shown that finite unions of n-dimensional tangent balls are not invariant under flat flows of any mean curvature flow with a bounded forcing. This is already proven in the two-dimensional case, so the third article generalizes this result to the higher dimensions.