Abstract

We investigate mean curvature flow of complete graphical hypersurfaces. These are complete hypersurfaces that can be written as graphs of functions defined on (bounded) domains and going to infinity at the boundary for completeness. We discuss two different curvature bounds for these flows, one that works below some arbitrary height and one that works above a cer- tain height and is dependent on the enveloping cylinder. We demonstrate what makes a uniform curvature bound difficult. Moreover, we give an example that behaves quite badly, illustrating issues that arise in this context. Lastly, we touch upon how one can obtain curvature bounds beyond singularities in the mean convex case.