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

TL;DR

This paper proves new topological restrictions on manifolds with positive scalar curvature using generalized soap bubbles, advancing understanding of scalar curvature constraints in differential geometry.

Contribution

It introduces the use of generalized soap bubbles to establish non-existence results for positive scalar curvature metrics on certain manifolds.

Findings

Closed aspherical 4- and 5-manifolds do not admit positive scalar curvature metrics.

Connected sums of tori with any manifold lack complete positive scalar curvature metrics for dimensions up to 7.

Supports the Schoen--Yau Liouville theorem for all locally conformally flat manifolds with non-negative 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 $n\leq 7$, 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).