On some properties of three-dimensional minimal sets in ${ℝ}^4$

Luu, Tien Duc

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 22 (2013) no. 3, pp. 465-493.

On prouve dans cet article la régularité Höldérienne pour les ensembles minimaux au sens d’Almgren de dimension 3 dans ${ℝ}^4$ autour d’un point de type $𝕐$ et dans le cas d’un ensemble Mumford-Shah minimal dans ${ℝ}^4$ qui est très proche d’un $𝕋$, l’existence d’un point avec une densité particulière. Cela donne une description locale des ensembles minimaux de dimension 3 dans ${ℝ}^4$ autour d’un point singulier et une propriété des ensembles Mumford-Shah minimaux dans ${ℝ}^4$.

We prove in this paper the Hölder regularity of Almgren minimal sets of dimension 3 in ${ℝ}^4$ around a $𝕐$-point and the existence of a point of particular type of a Mumford-Shah minimal set in ${ℝ}^4$, which is very close to a $𝕋$. This will give a local description of minimal sets of dimension 3 in ${ℝ}^4$ around a singular point and a property of Mumford-Shah minimal sets in ${ℝ}^4$.

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