Abstract

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 nonvanishing of certain index-theoretic invariants that arise at the infinity of the given underlying manifold. This is a ${\underleftarrow{\lim <!--FUNCTION\; APPLICATION-->}}^1$ phenomenon and naturally leads one to conjecture that Gromov’s compactness question has a positive answer provided that these ${\underleftarrow{\lim <!--FUNCTION\; APPLICATION-->}}^1$ invariants also vanish. We prove this is indeed the case for a class of $1$-tame manifolds.

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