On Gromov's compactness question regarding positive scalar curvature

TL;DR

This paper explores Gromov's compactness question for positive scalar curvature metrics on noncompact manifolds, providing both counterexamples and conditions under which the question has a positive answer.

Contribution

It constructs examples that negatively answer Gromov's question and proves positive results for certain classes of manifolds based on index invariants.

Findings

Counterexamples based on index invariants at infinity

Gromov's question has a positive answer when certain invariants vanish

Proves positive results for 1-tame manifolds

Abstract

In this paper, we give both positive and negative answers to Gromov's compactness question regarding positive scalar curvature metrics on noncompact manifolds. First we construct examples that give a negative answer to Gromov's compactness question. These examples are based on the non-vanishing of certain index theoretic invariants that arise at the infinity of the given underlying manifold. This is a $ \sideset{}{^1}\varprojlim$ phenomenon and naturally leads one to conjecture that Gromov's compactness question has a positive answer provided that these $ \sideset{}{^1}\varprojlim$ invariants also vanish. We prove this is indeed the case for a class of $1$-tame manifolds.