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Using minimal hypersurfaces, we obtain topological obstructions to admittingcomplete metrics with positive scalar curvature on a given class of non-compactn-manifolds with n less than 8. We show that the Liouville theorem for alocally conformally flat n-manifold of non-negative scalar curvature followsfrom the impossibility of there being a positive scalar curvature metric on itsconnect sum with the n-torus. With the recent work of Chodosh-Li, the Liouvilletheorem is now proved in all remaining cases. Finally, using MOTS instead ofminimal hypersurfaces, we show an Initial Data Set version of these resultswith the Dominant Energy Scalar appearing instead of positive scalar curvature.

Positive Scalar Curvature on Noncompact Manifolds and the Liouville Theorem