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

By Otis Chodosh. Annals of Mathematics, 2020

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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 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 $\mu$-bubbles).

Subfields: Applied Mathematics, Mathematical Physics, Nuclear and High Energy Physics

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