The isoperimetric profile of a compact Riemannian Manifold for small volumes
Abstract
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.
 Publication:

arXiv eprints
 Pub Date:
 October 2007
 arXiv:
 arXiv:0710.1396
 Bibcode:
 2007arXiv0710.1396N
 Keywords:

 Mathematics  Differential Geometry;
 53C21;
 58J60;
 5899;
 2899;
 4999
 EPrint:
 43 pages