The way space curves around itself is preserved even when a 3D shape is stretched or collapsed into a completely different form.
Scalar curvature lower bounds remain consistent when smooth three-manifolds converge under specific conditions. This discovery answers a famous open question about the stability of space when it undergoes extreme deformation. Mathematicians have struggled for years to prove that the 'curviness' of a shape doesn't just vanish during a collapse. This proof provides a more stable foundation for general relativity and our understanding of the geometry of the universe. It means that the fundamental structure of space-time is more resilient to change than previously thought. This stability is what allows the universe to maintain its physical laws even during violent cosmic events.
Scalar curvature under weak limits of manifolds
arXiv · 2605.03136
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,