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 $n\le 7$, the connected sum of a $n$-torus with an arbitrary manifold does not admit a complete metric of positive scalar curvature. When combined with 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).

Final version. Typos corrected, 5 figures added, several arguments clarified. To appear in Ann. Math

References in corpus (2)

• On the Multiplicity One Conjecture in Min-max theory
• Rigidity results for complete manifolds with nonnegative scalar curvature

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