Positive scalar curvature with point singularities

TL;DR

The paper investigates whether $L^\infty$-metrics can witness positive scalar curvature obstructions on manifolds that do not admit smooth psc metrics. It constructs high-codimension conical singularities on closed, simply connected spin manifolds via surgery to produce $L^\infty$-metrics that are smooth away from a singular set and have positive scalar curvature on the regular part, providing counterexamples to Schoen's conjecture for all $n\ge 8$ (including single-point singularities in certain $\mod 8$ cases). It also shows non-smoothability results on $\mathbb{R}^n$: there exist $L^\infty$-metrics with a point singularity that cannot be approximated by smooth metrics with $\mathrm{scal}\ge0$, highlighting limits of smoothing via Ricci-DeTurck flow. Finally, a KO-theoretic obstruction is established: if $j(\alpha(M;1))\neq\alpha(M;\Gamma)$ for a closed spin manifold $M$, then no such $L^\infty$-metric exists; enlargeable spin manifolds exemplify this obstruction. Together, the results separate global topological obstructions from local smoothing phenomena in scalar curvature under low-regularity metrics and connect index theory to singular geometric structures.

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.

Figures (2)

• Figure 1: The auxiliary function $\alpha\colon\mathbb{R}\to [0,1]$ and the warping function $f\colon \mathbb{R}\to (-\infty,1]$.
• Figure 2: The metric $\hat{g}$ on $M$: On $M\setminus \mathrm{B}^n$ we have a psc metric $g$ which is collared, then there is a transition region $\mathrm{S}^{n-1}\times [-R+\varepsilon,0]$ and on $\mathrm{S}^{n-1}\times(-R,R+\varepsilon]$ the metric $g_{\mathrm{cone}}$ is the linear scaling of $h$.

Theorems & Definitions (37)

• Definition 1.1: cf. CM:Skeleton
• Conjecture 1.2: Schoen, cf. CM:Skeleton
• Theorem 1
• Remark 1.3
• Remark 1.4
• Example 1.5: see \ref{['thm:exotic-spheres-have-l-infty-psc', 'remark:existence_of_non-trivial_Hitchin']}
• Example 1.6: see \ref{['thm:exotic-tori-have-l-infty-flat']}
• Remark 1.7: see \ref{['rem:Novikov']}
• Remark 1.8
• Theorem 2