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

We prove that for n ∈ {4, 5}, a closed aspherical n-manifold does not admit a Riemannian metric with positive scalar curvature. Additionally, we show that for n ≤ 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 µ-bubbles).