  In this work we consider a question in the calculus of variations motivated
by riemannian geometry, the isoperimetric problem. We show that solutions to
the isoperimetric problem, close in the flat norm to a smooth submanifold, are
themselves smooth and $C^{2,\alpha}$-close to the given sub manifold. We show
also a version with variable metric on the manifold. The techniques used are,
among other, the standards outils of linear elliptic analysis and comparison
theorems of riemannian geometry, Allard's regularity theorem for minimizing
varifolds, the isometric immersion theorem of Nash and a parametric version due
to Gromov.

Differential Geometry;Analysis of PDEs;Metric Geometry