Axially symmetric volume preserving mean curvature flow

In this thesis we study two problems of axially symmetric volume preserving mean curvature flow. We study the singularities of the flow for the major part of the thesis. We consider an axially symmetric hypersurface contained between the two parallel planes, meeting the planes at right angles along its boundary. We study the first singularity that develop in this flow. We prove that given a certain lower height bound on the boundary of a specific region, the first singularity is of type I. As part of the methodology used in this thesis we also derive extension theorems: we prove that no singularities can develop during a finite time interval, if the mean curvature is bounded within that time interval on the entire hypersurface. Finally we include new convergence results: Assuming the surface is not pinching off along the axis at any time during the flow, and without any additional conditions, as for example on the curvature, we prove that it converges to a hemisphere, when the hypersurface has a free boundary and satisfies Neumann boundary data, and to a sphere when it is compact without boundary.