abstract. For a complete noncompact Riemannian manifold with smoothly bounded geometry, we prove the existence of isoperimetric regions in a larger space obtained by adding finitely many limit manifolds at infinity. As one of many possible applications, we extend properties of the isoperimetric profile from compact manifolds to such noncompact manifolds.

We define a new class of submanifolds called pseudo-bubbles, defined by an equation weaker than constancy of mean curvature. We show that in a neighborhood of each point of a Riemannian manifold, there is a unique family of concentric pseudo-bubbles which contains all the pseudo-bubbles C 2,α -close to small spheres. This permit us to reduce the isoperimetric problem for small volumes to a variational problem in finite dimension.

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We provide an isoperimetric comparison theorem for small volumes in an n-dimensional Riemannian manifold (M n , g) with strong bounded geometry, as in Definition 2.3, involving the scalar curvature function. Namely in strong bounded geometry, if the supremum of scalar curvature function S g < n(n − 1)k 0 for some k 0 ∈ R, then for small volumes the isoperimetric profile of (M n , g) is less then or equal to the isoperimetric profile of M n k 0 the complete simply connected space form of constant sectional curvature k 0 . This work generalizes Theorem 2 of [Dru02b] in which the same result was proved in the case where (M n , g) is assumed to be just compact. As a consequence of our result we give an asymptotic expansion in Puiseux's series up to the second nontrivial term of the isoperimetric profile function for small volumes. Finally, as a corollary of our isoperimetric comparison result, it is shown, in the special case of manifolds with strong bounded geometry, and S g < n(n − 1)k 0 that for small volumes the Aubin-Cartan-Hadamard's Conjecture in any dimension n is true.