Fully nonlinear curvature flow of axially symmetric hypersurfaces

In this thesis we consider axially symmetric evolving hypersurfaces mostly with boundary conditions between two parallel planes. The speed function is a fully nonlinear function of the principal curvatures of the hypersurface, homogeneous of degree one. We have results for several boundary conditions. Specifically, with a natural class of Neumann boundary conditions we show that evolving hypersurfaces exist for a finite maximal time. The maximal time is characterised by a curvature singularity at either boundary. Generally, the singularities of the flow are classified as Type I in the case of pure Neumann boundary conditions. In addition to the "curvature pinching estimate" that is obtained, Sturmian theory is applied to show the discreteness of singularities. Furthermore, some results carry over to higher degrees of homogeneity. Finally, we have some additional results including a gradient bound for the hight function in the volume preserve case.