The isoperimetric profile of a compact Riemannian Manifold for small volumes

Stefano Nardulli (1, 2)

1 Dipartimento di Metodi et Modelli Matematici
2 LM-Orsay - Laboratoire de Mathématiques d'Orsay

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 allows us to reduce the isoperimetric problem for small volumes to a variational problem in finite dimension.

Mots clés: regularity theory; isoperimetric profile; riemannian geometry; geometric measure theory; PDE's methods; regularity theory.

Domaines: Géométrie différentielle [math.DG]

Stefano Nardulli. The isoperimetric profile of a compact Riemannian Manifold for small volumes. 2007. ⟨hal-00177259⟩