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 $n\le 7$, 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).
Citation
Otis Chodosh. Chao Li. "Generalized soap bubbles and the topology of manifolds with positive scalar curvature." Ann. of Math. (2) 199 (2) 707 - 740, March 2024. https://doi.org/10.4007/annals.2024.199.2.3
Information