Volume growth functions of complete Riemannian manifolds with positive scalar curvature

TL;DR

This paper investigates the volume growth of complete Riemannian manifolds with positive scalar curvature, demonstrating that certain infinite connected sum manifolds can admit metrics with prescribed volume growth types.

Contribution

It proves that manifolds formed by infinite connected sums of positive scalar curvature manifolds can support metrics with volume growth matching a given bounded derivative growth function.

Findings

Positive scalar curvature metrics can be constructed with prescribed volume growth.

Infinite connected sums of certain manifolds admit metrics with controlled volume growth.

The results extend to manifolds formed by sums along lower-dimensional spheres.

Abstract

Let $M$ be an open manifold of dimension at least $3$, which admits a complete metric of positive scalar curvature. For a function $v$ with bounded growth of derivative, whether $M$ admits a metric of positive scalar curvature with volume growth of the same growth type as $v$ is unknown. We answer this question positively in the case of manifolds, which are infinite connected sums of closed manifolds that admit metrics of positive scalar curvature. To define a metric of positive scalar curvature with a certain volume growth type on $M$, we use the Gromov-Lawson construction of metrics with positive scalar curvature on connected sums and Grimaldi-Pansu's construction of metrics of bounded geometry of certain volume growth type on open manifolds. We generalize this result to manifolds, which are infinite connected sums of similar closed manifolds along lower-dimensional spheres.