Differentiability properties of the isoperimetric profile in complete noncompact Riemannian manifolds with $$C^0$$-locally asymptotic bounded geometry

Abstract

For a complete noncompact connected Riemannian manifold with bounded geometry M n , we prove that the isoperimetric profile function I M n is twice differentiable almost everywhere. Moreover, we show that a differential inequality is satisfied by I M ; extending in this way well-known results for compact manifolds due to Bavard and Pansu (Ann Sci École Norm Sup :479-490, 1986), to this class of noncompact complete Riemannian manifolds with bounded geometry. Here for C 0-locally asymptotic bounded geometry we mean that for all pointed sequences p j ∈ M diverging at infinity the sequence of pointed Riemannian manifolds (M, p j , g) sub-converge in C 0 topology to a limit manifold (M ∞ , g ∞ , p ∞) that we assume to be at least of class C 2 .