Calculus of Variations and Geometric Measure Theory

S. Nardulli - L. E. Osorio Acevedo

Sharp isoperimetric inequalities for small volumes in complete noncompact Riemannian manifolds of bounded geometry involving the scalar curvature

created by nardulli on 05 Nov 2016
modified on 14 Jul 2020


Published Paper

Inserted: 5 nov 2016
Last Updated: 14 jul 2020

Journal: International Mathematics Research Notices
Pages: 72
Year: 2016
Doi: 10.1093/imrn/rny131


We provide an isoperimetric comparison theorem for small volumes in an n-dimensional Riemannian manifold (Mn,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 Sg < n(n − 1)k0 for some k0 ∈ R, then for small volumes the isoperimetric profile of (Mn,g) is less then or equal to the isoperimetric profile of Mnk0 the complete simply connected space form of constant sectional curvature k0. This work generalizes Theorem 2 of Dru02b in which the same result was proved in the case where (M,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 Sg < n(n − 1)k0 that for small volumes the Aubin-Cartan-Hadamard’s Conjecture in any dimension n is true.