On the boundary of the almost isoperimetric domains

Erwann Aubry (1) , Jean-Franç Ois Grosjean (2)

1 Géométrie et Analyse
2 IECL - Institut Élie Cartan de Lorraine

We prove that finite perimeter subsets of R n+1 with small isoperimetric deficit have boundary Hausdorff-close to a sphere up to a subset of small measure. We also refine this closeness under some additional a priori integral curvature bounds. As an application, we answer a question raised by B. Colbois concerning the almost extremal hypersurfaces for Chavel's inequality.

Domaines: Géométrie différentielle [math.DG]

Erwann Aubry, Jean-Franç Ois Grosjean. On the boundary of the almost isoperimetric domains. 2017. ⟨hal-01489270⟩