Positive Scalar Curvature on Noncompact Manifolds and the Liouville Theorem

Abstract

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

arXiv commentComments are welcome

References in corpus (5)

• The Higher Dimensional Positive Mass Theorem II
• Generalized soap bubbles and the topology of manifolds with positive scalar curvature
• No metrics with Positive Scalar Curvatures on Aspherical 5-Manifolds
• Rigidity results for complete manifolds with nonnegative scalar curvature
• Contractible $3$-manifold and Positive scalar curvature (I)

Cited by in corpus (2)

• The mass of an asymptotically hyperbolic end and distance estimates
• The positive mass theorem and distance estimates in the spin setting