$C^{1+α}$-Regularity for Two-Dimensional Almost-Minimal Sets in $\R^n$

Details $C^{1+α}$-Regularity for Two-Dimensional Almost-Minimal Sets in $\R^n$ $C^{1+α}$-Regularity for Two-Dimensional Almost-Minimal Sets in $\R^n$

2008-06-12

arxiv.org/abs/0806.2080v1

Mathematics

Classical Analysis and ODEs

115 pages, 4 figures

Scientific paper

We give a new proof and a partial generalization of Jean Taylor's result [Ta] that says that Almgren almost-minimal sets of dimension 2 in $\R^3$ are locally $C^{1+\alpha}$-equivalent to minimal cones. The proof is rather elementary, but uses a local separation result proved in [D3] and an extension of Reifenberg's parameterization theorem [DDT]. The key idea is still that if $X$ is the cone over an arc of small Lipschitz graph in the unit sphere, but $X$ is not contained in a disk, we can use the graph of a harmonic function to deform $X$ and diminish substantially its area. The local separation result is used to reduce to unions of cones over arcs of Lipschitz graphs. A good part of the proof extends to minimal sets of dimension 2 in $\R^n$, but in this setting our final regularity result on $E$ may depend on the list of minimal cones obtained as blow-up limits of $E$ at a point.

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