Scalar curvature under weak limits of manifolds

Abstract

We show that scalar curvature lower bounds are preserved under certain weak convergence of smooth three manifolds to a smooth limit. More precisely, suppose that $M_k$ and $M$ are smooth, closed, Riemannian three manifolds. Assume that there are smooth, surjective, $\lambda_k$-Lipschitz maps $f_k\colon M_k \to M$ and that $\text{Vol}(M_k)\to \text{Vol}(M)$ and $\lambda_k\to 1$. Then if each $M_k$ has scalar curvature bounded below by $\kappa$ so does $M$. This result answers questions of Gromov, Sormani, Allen, and others. The proof relies on a delicate comparison between $\mu$-bubbles in $M_k$ and $\mu$-bubbles in $M$.