Curvature Flows on Polygons and Surface Clusters

Abstract

This thesis explores geometric variational problems on various contexts. We consider the evolution of planar polygons under the flow which decrease the L^p-norm of a polygon. We derive a natural generalization of Huisken's monotonicity formula for this flow. As a direct corollary, we prove that the solutions to this flow converge asymptotically to self-similar solutions. We investigate the motion of surface clusters under the volume preserving mean curvature flow. We show the short time existence of this flow on n-dimensional piecewise regular, compact hypersurface clusters in parabolic Hölder space. We then consider the planar double bubble evolving by this flow. We derive some integral estimates for the curvature, and, as a consequence, we prove that a double bubble converges to the standard one.