Some constructions of uniformly positive scalar curvature metrics on open manifolds

Abstract

We obtain several constructions of uniformly positive scalar curvature complete Riemannian metrics on open manifolds. For dimension $n\ge 3$, we show that if such a manifold admits a proper Morse function $f$ bounded below such that $f$ has no critical points of index $\ge n-2$, then it admits a uniformly positive scalar curvature metric. On the other hand if such a manifold admits a positive scalar curvature metric along with a compact exhaustion $\{U_i\}$ such that the boundary of each $U_i$ is minimal, then it also admits a uniformly positive scalar curvature metric. For dimension $4\le n\le 7$, we show that if the manifold has product ends and a positive scalar curvature metric with $C$-quadratic decay at infinity for $C>4{\pi }^2$ with respect to some basepoint, then the existence of a mean convex hypersurface far enough from the basepoint implies the existence of a uniformly positive scalar curvature metric on the manifold. We study some applications of these results, including showing that if an open manifold of dimension $n\ge 3$ that admits no uniformly positive scalar curvature metric has a positive scalar curvature metric with mean convex exhaustion, then it admits a mean convex foliation of compact sets sufficiently close to the ends. On the other hand, if such a manifold has a mean concave exhaustion, then its ends admit a mean concave foliation.

18 pages, 3 figures