Positive scalar curvature with point singularities

TL;DR

This paper constructs high-dimensional manifolds with singular metrics that have positive scalar curvature outside a singular set, challenging previous conjectures and highlighting differences between singular and smooth metrics.

Contribution

It provides counterexamples to Schoen's conjecture by constructing manifolds with singular metrics of positive scalar curvature that cannot be approximated smoothly, and establishes a KO-theoretic obstruction for certain metrics.

Findings

Existence of manifolds with singular psc metrics but no smooth psc metrics.

Counterexamples to smoothing singular metrics of psc.

KO-theoretic obstructions for certain singular metrics.

Abstract

We show that in every dimension $n \geq 8$, there exists a smooth closed manifold $M^n$ which does not admit a smooth positive scalar curvature ("psc") metric, but $M$ admits an $\mathrm{L}^\infty$-metric which is smooth and has psc outside a singular set of codimension $\geq 8$. This provides counterexamples to a conjecture of Schoen. In fact, there are such examples of arbitrarily high dimension with only single point singularities. We also discuss related phenomena on exotic spheres and tori. In addition, we provide examples of $\mathrm{L}^\infty$-metrics on $\mathbb{R}^n$ for certain $n \geq 8$ which are smooth and have psc outside the origin, but cannot be smoothly approximated away from the origin by everywhere smooth Riemannian metrics of non-negative scalar curvature. This stands in precise contrast to established smoothing results via Ricci-DeTurck flow for singular metrics with stronger regularity assumptions. Finally, as a positive result, we describe a $\mathrm{KO}$-theoretic condition which obstructs the existence of $\mathrm{L}^\infty$-metrics that are smooth and of psc outside a finite subset. This shows that closed enlargeable spin manifolds do not carry such metrics.