Title: Positive scalar curvature metrics and aspherical summands
Abstract: It has been a classical question which manifolds
admit Riemannian metrics with positive scalar curvature. A closed
manifold is called aspherical if it has contractible universal
cover. It has been conjectured since the 80s that all closed
aspherical manifolds do not admit metric with positive scalar
curvature. By works of Gromov-Lawson, Chodosh-Li, Gromov, and
Chodosh-Li-Liokumovich, for n = 3, 4, 5, if a closed n-manifold M
admits a map of nonzero degree to a closed aspherical n-manifold, then
M does not admit a metric with positive scalar curvature. We prove for
n = 3,4,5 that the connected sum of a closed aspherical n-manifold
with an arbitrary non-compact manifold does not admit a complete
metric with nonnegative scalar curvature. In particular, a special
case of our result answers a question of Gromov. More generally, for
n=3,4,5, we give a partial classification result of complete
n-manifolds with positive scalar curvature and prove that no complete
n-manifold with a non-zero degree map to a closed aspherical
n-manifold can have positive scalar curvature. This result confirms
the validity of Gromov's non-compact domination conjecture for closed
aspherical manifolds of dimensions 3, 4, and 5. This is joint work
with Jianchun Chu and Jintian Zhu.