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
phenomenon and naturally leads one to conjecture that Gromov’s
compactness question has a positive answer provided that these
invariants also vanish. We prove this is indeed the case for a class of
-tame
manifolds.
Keywords
Gromov's compactness conjecture on scalar curvature, index
invariants at infinity