Mean Curvature Flow with Triple Junctions in Higher Space Dimensions

Abstract

We consider mean curvature flow of n-dimensional surface clusters. At (n − 1)-dimensional triple junctions an angle condition is required which in the symmetric case reduces to the well-known 120 degree angle condition. Using a novel parametrization of evolving surface clusters and a new existence and regularity approach for parabolic equations on surface clusters we show local well-posedness by a contraction argument in parabolic Hölder spaces.

A local regularity theorem for mean curvature flow with triple edges

“…Since N i only has one triple junction in B r (0) × (0, min{T i , T }) and no closed loops, it implies that any tangent flow at (x, T i ), where x ∈ B r (0), is a smooth, self-similarly shrinking network without closed loops and at most one triple junction. Since N i is Y -regular, Lemma 7.1 together with (4) implies that N i is smooth in a neighbourhood of (x, T i ). Thus T i ≥ T .…”

“…Remark A. 4. It suffices to assume that u ∈ C 1 (U × [0, 1)) since Brakke's local regularity theorem [2] (or alternatively [18]) implies that u is smooth for t > 0.…”

On convergence of solutions to equilibria for fully nonlinear parabolic systems with nonlinear boundary conditions

“…Note that the operators G 1 and G 2 are completely local as the projection pr acts as the identity on its image ∂Γ * . Altogether, by recalling the parameterization (see (4.4)), we are led to the following nonlinear, nonlocal problem (see [10,Equation (20)] for the analogous result obtained for the mean curvature flow): 20) where the term ±l * should be understood in a sense that +l * is taken in (4.20) for the values of x in the neighborhood of l * and −l * is taken in (4.20) for the values of x in the neighborhood of −l * .…”

On the Surface Diffusion Flow with Triple Junctions in Higher Space Dimensions

“…In the literature, most existence results for geometric flows of higher dimensional surfaces only give analytic results. For example, Depner, Garcke and Kohsaka proved in [DGK14] analytic existence for triple junction clusters evolving due to mean curvature flow. Indeed, the strategy of our proof is based on this work.…”