Regularity of the Level Set Equation for Mean Curvature Flow with an Axis of Symmetry

Abstract

In this thesis we study the regularity of viscosity solutions to the level set equation for mean curvature flow. We describe a set of hypotheses under which we can prove that the level sets of these solutions are C¹,¹ submanifolds of spacetime with well understood behavior near singular times. We then relate the derivatives of the solution of the level set flow to the solutions of certain evolution equations along fixed level sets. Finally we carry out this program to show that certain solutions with an axis of symmetry are in fact classical solutions of the level set problem.