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

TL;DR

This paper establishes sharp isoperimetric inequalities for small volumes in noncompact Riemannian manifolds with bounded geometry, involving scalar curvature, and extends previous compact case results to a broader class of manifolds.

Contribution

It generalizes isoperimetric comparison theorems to noncompact manifolds with bounded geometry, involving scalar curvature, and provides asymptotic expansions of the isoperimetric profile for small volumes.

Findings

Isoperimetric profile of the manifold is bounded by that of a model space form for small volumes.

Derived asymptotic expansion of the isoperimetric profile function up to the second nontrivial term.

Confirmed the Aubin-Cartan-Hadamard conjecture for small volumes in manifolds with strong bounded geometry.

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 $k_0\in\mathbb{R}$, then for small volumes the isoperimetric profile of $(M^n,g)$ is less then or equal to the isoperimetric profile of $\mathbb{M}^n_{k_0}$ 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.