On regularity for mean curvature flow of hypersurfaces
A family of isometric immersions \(x_ t : M^ n \to \mathbb{R}^{n+1}\) of a parameter manifold \(M^ n\) into the Euclidean space \(\mathbb{R}^{n+1}\) is said to move by mean curvature if \({d \over dt} x_ t\) equals the mean curvature vector. It is known that the hypersurfaces \(M_ t = x_ t (M^ n)\) can become singular after a finite time interval \((0,T)\). The author surveys results on singularities or on regularity of his own as well as of others. The heat operator \({\partial \over \partial t} - \Delta\) of \(\mathbb{R}^{n+1}\) and the square \(| A |^ 2\) of the second fundamental form \(A = A_ t\) of \(M_ t\) are used to derive a priori estimates for the mean curvature flow. For instance, the supremum of \(| A |^ 2\) in some domain is bounded by the mean value of \(| A |^ 2\) in a somewhat bigger domain. Another remarkable result is that, under plausible assumptions, the singular set has Hausdorff measure zero.
scientific article; zbMATH DE number 719390
scientific article