We show that smooth isoperimetric profiles are exceptional for real analytic Riemannian manifolds. For instance, under some extra assumption, this can happen only on topological spheres. Stefano Nardulli visitará granada desde el 29 de abril al 7 de junio.

We show that, the solutions of the isoperimetric problem for small volumes are $C^{2,alpha }$-close to small spheres. On the way, we define a class of submanifolds called pseudo balls, defined by an equation weaker than constancy of mean curvature. We show that in a neighborhood of each point of a compact riemannian manifold, there is a unique family concentric pseudo balls which contains all the pseudo balls $C^{2,alpha }$-close to small spheres. This permit us to reduce the isoperimetric problem for small volumes to a variational problem in finite dimension. Stefano Nardulli visitará Granada desde el 29 de abril al 7 de junio.