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 Ecole 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 $$p_j\in M$$ diverging at infinity the sequence of pointed Riemannian manifolds $$(M,p_j,g)$$ sub-converge in $$C^0$$ topology to a limit manifold $$(M_\infty , g_\infty , p_\infty )$$ that we assume to be at least of class $$C^2$$ .