Singular metrics with nonnegative scalar curvature and RCD

TL;DR

This paper proves that certain singular metrics with nonnegative scalar curvature on specific manifolds are Ricci flat and extend smoothly over singularities, confirming Schoen's Conjecture in these cases.

Contribution

It demonstrates that such metrics are Ricci flat and smooth across singularities, establishing the space as an RCD(0, n) space, thus confirming Schoen's Conjecture for these cases.

Findings

Metrics are Ricci flat and extend smoothly over singularities.

The space has nonnegative synthetic Ricci curvature (RCD(0, n)).

Results hold for singular sets of finite unions of submanifolds under certain conditions.

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.