A local regularity theorem for mean curvature flow with triple edges. (English) Zbl 1431.53101
Summary: Mean curvature flow of clusters of \(n\)-dimensional surfaces in \(\mathbb{R}^{n+k}\) that meet in triples at equal angles along smooth edges and higher order junctions on lower-dimensional faces is a natural extension of classical mean curvature flow. We call such a flow a mean curvature flow with triple edges. We show that if a smooth mean curvature flow with triple edges is weakly close to a static union of three \(n\)-dimensional unit density half-planes, then it is smoothly close. Extending the regularity result to a class of integral Brakke flows, we show that this implies smooth short-time existence of the flow starting from an initial surface cluster that has triple edges, but no higher order junctions.
MSC:
| 53E10 | Flows related to mean curvature |
| 53A07 | Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces |
| 53C42 | Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) |
| 49Q20 | Variational problems in a geometric measure-theoretic setting |
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