Size minimizing surfaces

Pauw, Thierry De

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 42 (2009) no. 1, pp. 37-101.

Nous obtenons un nouveau théorème d’existence relatif au problème de Plateau dans l’espace euclidien de dimension $3$. Ce faisant, nous comparons les approches d’E.R. Reifenberg d’une part, et de H. Federer et W.H. Fleming d’autre part. Un pas technique important consiste à démontrer qu’on peut approcher tout ensemble compact et rectifiable, en mesure de Hausdorff et en distance de Hausdorff, par une surface localement acyclique ayant le même bord.

We prove a new existence theorem pertaining to the Plateau problem in $3$-dimensional Euclidean space. We compare the approach of E.R. Reifenberg with that of H. Federer and W.H. Fleming. A relevant technical step consists in showing that compact rectifiable surfaces are approximatable in Hausdorff measure and in Hausdorff distance by locally acyclic surfaces having the same boundary.

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