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 -bubbles).