Level set approach to mean curvature flow in arbitrary codimension (Q1126421)

The authors develop a level set theory for the mean curvature evolution of surfaces with arbitrary codimension which generalizes the work of \textit{L. C. Evans} and \textit{J. Spruck} [J. Differ. Geom. 33, No. 3, 635-681 (1991; Zbl 0726.53029)] and \textit{Y.-G. Chen}, \textit{Y. Giga} and \textit{S. Goto} [J. Differ. Geom. 33, 749-786 (1991; Zbl 0715.35037)] on the codimension one case. The main idea is to surround the evolving surface of codimension \(k\) in \(\mathbb{R}^d\) 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. This is motivated by the theory of barriers proposed by \textit{E. De Giorgi} [Barriers, boundaries, motion of manifolds, Lectures held in Pavia, Italy (1994)] to describe mean curvature flow in arbitrary codimensions. The authors show that the level set solutions agree with the classical solutions whenever these exist, and that the varifold solutions of \textit{K. A. Brakke} [The motion of a surface by its mean curvature, Princeton University Press, Princeton, N.J. (1978; Zbl 0386.53047)] are included in the level set solutions.