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

Content Provider	Springer Nature Link
Author	David, Guy
Copyright Year	2010
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 ℝ$^{3}$ 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 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.
Starting Page	837
Ending Page	954
Page Count	118
File Format	PDF
ISSN	10506926
Journal	Journal of Geometric Analysis
Volume Number	20
Issue Number	4
e-ISSN	1559002X
Language	English
Publisher	Springer-Verlag
Publisher Date	2010-05-01
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Minimal sets Almgren restricted or quasiminimal sets Epiperimetric inequalities Hausdorff measure Variational problems in a geometric measure-theoretic setting Global Analysis and Analysis on Manifolds Dynamical Systems and Ergodic Theory Abstract Harmonic Analysis Fourier Analysis Convex and Discrete Geometry Differential Geometry
Content Type	Text
Resource Type	Article
Subject	Geometry and Topology

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