Existence of Positive Scalar Curvature and Positive Yamabe constant on Hypersurfaces of Noncompact Cylinders

Abstract

Let $ X $ be an oriented, closed manifold with $ \dim X \geqslant 2 $. Let $ (Z, \partial Z) $ be an oriented, compact manifold with (possibly empty) smooth boundary and $ \dim Z \geqslant 2 $. In this article, we show that if the noncompact cylinder $ X \times \mathbb{R} $ admits a complete Riemannian metric $ g $ with positive injectivity radius and uniformly positive scalar curvature, and that is of bounded geometry or bounded curvature, then $ X $ admits a positive scalar curvature metric within the same conformal class provided that some $ g $-angle condition is satisfied. This partially answers a conjecture of Rosenberg and Stolz \cite{RosSto} without topological assumptions. With the $ g $-angle condition, we can also show that if $ (Z \times \mathbb{R}, \partial Z \times \mathbb{R}) $ admits a complete metric $ g $ that has positive Yamabe constant and positive injectivity radius, and is of bounded geometry or bounded curvature, then $ (Z, \partial Z) $ has positive Yamabe constant for the conformal class $ [\imath^{*}g] $ with the natural inclusion $ \imath : (Z, \partial Z) \hookrightarrow (Z \times \mathbb{R}, \partial Z \times \mathbb{R}) $.