Generalized soap bubbles and the topology of manifolds with positive scalar curvature

Otis Chodosh, Chao Li

Published 2020-08-27Version 1

We prove that a closed aspherical $4$-manifold does not admit a Riemannian metric with positive scalar curvature. Additionally, we show that for $n\leq 7$, the connected sum of a $n$-torus with an arbitrary manifold does not admit a complete metric of positive scalar curvature. When combined with forthcoming contributions by Lesourd--Unger--Yau, this proves that the Schoen--Yau Liouville theorem holds for all locally conformally flat manifolds with non-negative scalar curvature. A key tool in these results are generalized soap bubbles---surfaces that are stationary for prescribed-mean-curvature functionals (also called as $\mu$-bubbles).