Level set approach to mean curvature flow in arbitrary codimension

We develop a level set theory for the mean curvature evolution of surfaces with arbitrary co-dimension, thus generalizing the previous work [6, 13] on hypersurfaces. The main idea is to surround the evolving surface of co-dimension k in R by a family of hypersurfaces (the level sets of a function) evolving with normal velocity equal to the sum of the (d − k) smallest principal curvatures. The existence and the uniqueness of a weak (level-set) solution, is easily established using mainly the results of [6] and the theory of viscosity solutions for second order nonlinear parabolic equations. The level set solutions coincide with the classical solutions whenever the latter exist. The proof of this connection uses a careful analysis of the squared distance from the surfaces. It is also shown that varifold solutions constructed by Brakke [5] are included in the level-set solutions. The idea of surrounding the evolving surface by a family of hypersurfaces with a certain property is related to the barriers of DeGiorgi. An introduction to the barriers and their connection to the level set solutions is also provided.