Scalar curvature under weak limits of manifolds

TL;DR

The paper proves that scalar curvature lower bounds are maintained under certain weak limits of smooth three manifolds, using a comparison of $ ext{mu}$-bubbles.

Contribution

It establishes scalar curvature preservation under weak convergence with Lipschitz maps and volume convergence, answering longstanding questions in geometric analysis.

Findings

Scalar curvature bounds are preserved under specified weak limits.

The proof uses comparison of $ ext{mu}$-bubbles in manifolds and their limits.

Addresses open questions posed by Gromov, Sormani, and Allen.

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 $M$.