LEVEL SET APPROACH TO MEAN-CURVATURE FLOW IN ARBITRARY CODIMENSION

We develop a level set theory for the mean curvature evolution of surf aces with arbitrary co-dimension, thus generalizing the previous work [8, 15] on hypersurfaces. The main idea is to surround the evolving su rface of codimension-k in R(d) 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 uniq ueness of a weak (level-set) solution is easily established by using m ainly the results of [8] and the theory of viscosity solutions for sec ond order nonlinear parabolic equations. The level set solutions coinc ide with the classical solutions whenever the latter exist. The proof of this connection uses a careful analysis of the squared distance fro m the surfaces. It is also shown that varifold solutions constructed b y Brakke [7] are included in the level-set solutions. The idea of surr ounding the evolving surface by a family of hypersurfaces with a certa in property is related to the barriers of De Giorgi. An introduction t o the theory of barriers and its connection to the level set solutions is also provided.