Geometric evolution problems, distance function and viscosity solutions

Abstract

The mean curvature flow is a geometric initial value problem. Starting from a smooth initial surface Γ₀in Rⁿ, the surfacesΓ_tevolve in time with normal velocity equal to their mean curvature vector. By parametric methods of differential geometry many results have been obtained for convex surfaces, graphs or planar curves (see for instance Altschuler & Grayson [AG92], Ecker & Huisken [EH89], Gage & Hamilton [GH86], Grayson [Gra87], and Huisken [Hui84]). However, for n ≥ 3, initially smooth surfaces may develop singularities. For example, a “dumbbell” region in R3 splits into two pieces in finite time (cf. [Gra89a]) or a “fat” enough torus closes its interior hole in finite time (cf. [SS93]). Also it can be seen that smooth curves in R³may self intersect in finite time.