Global regularity for minimal sets near a union of two planes

Liang, Xiangyu

Annales de l'Institut Fourier, Tome 66 (2016) no. 5, pp. 2067-2099.

On traite la régularité globale des ensembles minimaux 2-dimensionnels qui sont proches d’une union de deux plans, et on démontre que tout ensemble minimal proche d’une union de deux plans presque orthogonaux à l’infini dans ${ℝ}^4$ est un cône. L’enjeu est de contrôler le comportement d’un ensemble minimal à petite échelle par la topologie à grande échelle.

We discuss the global regularity of 2 dimensional minimal sets that are near a union of two planes, and prove that every global minimal set in ${ℝ}^4$ that looks like a union of two almost orthogonal planes at infinity is a cone. The main point is to use the topological properties of a minimal set at a large scale to control its behavior at smaller scales.

Classification : 28A75, 49Q10, 49Q20, 49K99
Keywords: Minimal sets, blow-in limit, existence of singularities, Hausdorff measure, elliptic systems
Mot clés : Ensembles minimaux, limites d’explosion, existence de singularités, mesure de Hausdorff, système elliptiques.

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