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

Abstract

We prove that for $n\in \{4,5\}$, a closed aspherical $n$-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 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 geometric tool in these results are generalized soap bubbles—surfaces that are stationary for prescribed-mean-curvature functionals (also called $\mu $-bubbles).