Abstract
We give a new proof and a partial generalization of Jean Taylor’s result (Ann. Math. (2) 103(3), 489–539, 1976) that says that Almgren almost-minimal sets of dimension 2 in ℝ3 are locally C 1+α-equivalent to minimal cones. The proof is rather elementary, but uses a local separation result proved in Ann. Fac. Sci. Toulouse 18(1), 65–246, 2009 and an extension of Reifenberg’s parameterization theorem (David et al. in Geom. Funct. Anal. 18, 1168–1235, 2008). 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 substantially diminish 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 ℝn, 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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David, G. C1+α-Regularity for Two-Dimensional Almost-Minimal Sets in ℝn. J Geom Anal 20, 837–954 (2010). https://doi.org/10.1007/s12220-010-9138-z
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DOI: https://doi.org/10.1007/s12220-010-9138-z