Sharp Isoperimetric Inequalities for Small Volumes in Complete Noncompact Riemannian Manifolds of Bounded Geometry Involving the Scalar Curvature

Abstract

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.

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