Positive scalar curvature on aspherical compact manifolds

Prove that no closed aspherical smooth manifold admits a Riemannian metric of positive scalar curvature.

Background

The paper begins by recalling a longstanding problem at the intersection of geometry and topology concerning the existence of positive scalar curvature metrics on aspherical compact manifolds. This conjecture has been approached via minimal hypersurface methods and via Dirac operator techniques, but remains unresolved in general.

The authors situate their noncompact results against this classical compact-manifold conjecture, highlighting the broader context in which index-theoretic obstructions to positive scalar curvature are studied.