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

Abstract

Let $X$ be an oriented, closed manifold with $\dim X⩾2$. Let $(Z,\partial Z)$ be an oriented, compact manifold with (possibly empty) smooth boundary and $\dim Z⩾2$. 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 $[{ı}^{\ast }g]$ with the natural inclusion $ı:(Z,\partial Z)\hookrightarrow (Z\times \mathbb{R},\partial Z\times \mathbb{R})$.

23 Pages, all comments are welcome