The isoperimetric profile of a smooth Riemannian manifold for small volumes

We define a new class of submanifolds called pseudo-bubbles, defined by an equation weaker than constancy of mean curvature. We show that in a neighborhood of each point of a Riemannian manifold, there is a unique family of concentric pseudo-bubbles which contains all the pseudo-bubbles C ^2,α -close to small spheres. This permit us to reduce the isoperimetric problem for small volumes to a variational problem in finite dimension. Work supported by a grant from INDAM “Istituto Nazionale di Alta Matematica Francesco Severi” (Roma).