Singular metrics with nonnegative scalar curvature and RCD
Xianzhe Dai, Changliang Wang, Lihe Wang, Guofang Wei
TL;DR
The paper addresses whether uniformly Euclidean metrics with nonnegative scalar curvature on closed manifolds admit removable singularities and Ricci-flat extensions, focusing on manifolds of the form $M^n = T^n \# M_0$ with $4 \le n \le 7$ or $M_0$ spin. The authors embed the problem in the synthetic Ricci curvature framework by proving such metrics yield ${\rm RCD}(0,n)$ spaces under suitable codimension and Ricci lower bounds, and then establish gradient and Hessian estimates for eigenfunctions to verify the BE$(K,N)$ condition. This leads to a global removable singularity result, and, via the splitting theorem for RCD spaces, to the rigidity conclusion that the smooth part is Ricci-flat and the metric extends smoothly (potentially after altering the differentiable structure), confirming Schoen's Conjecture in these cases. They further show that the result extends to singular sets that are finite unions of submanifolds intersecting transversally, highlighting a topologically robust route to removability. The work connects scalar curvature rigidity with the modern theory of metric measure spaces and provides a versatile toolkit for analyzing singular geometric structures with synthetic Ricci curvature notions.
Abstract
We show that a uniformly Euclidean metric with isolated singularity on $M^n = T^n \# M_0$, where $4\leq n\leq 7$ or $n\geq 4$, $M_0$ spin, and nonnegative scalar curvature on the smooth part is Ricci flat and extends smoothly over the singularity. This confirms Schoen's Conjecture in these cases. The key to the proof is to show that the space has nonnegative synthetic Ricci curvature, i.e., an $RCD(0, n)$ space. Our result also holds when the singular set consists of a finite union of submanifolds (of possibly different dimensions) intersecting transversally under additional assumption on the co-dimension and the location of the singular set.