Desingularizing positive scalar curvature 4-manifolds

Abstract

We show that the bordism group of closed 3-manifolds with positive scalar curvature (psc) metrics is trivial by explicit methods. Our constructions are derived from scalar-flat Kähler ALE surfaces discovered by Lock-Viaclovsky. Next, we study psc 4-manifolds with metric singularities along points and embedded circles. Our psc null-bordisms are essential tools in a desingularization process developed by Li-Mantoulidis. This allows us to prove a non-existence result for singular psc metrics on enlargeable 4-manifolds with uniformly Euclidean geometry. As a consequence, we obtain a positive mass theorem for asymptotically flat 4-manifolds with non-negative scalar curvature and low regularity.

Appendix A

In this section we will provide some details on the minimization problem found in Claim 1 during the proof of Theorem A. The argument we present here is only a slight modification of the proof of [28, Lemma 6.1].

Some notation: if (M, g) is a Riemannian manifold and k is some whole number, we write ${\mathcal {H}}_g^{k}$ for the k-dimensional Hausdorff measure associated with g.

Lemma 2

Suppose $g'$ is a smooth metric on ${\mathbb {R}}^2\times S^2$ which satisfies

$$\begin{aligned} \lambda ^{-1}(g_{{\mathbb {H}}^2}+g_{S^2})\le g'\le \lambda (g_{{\mathbb {H}}^2}+g_{S^2}) \end{aligned}$$

(26)

for a positive constant $\lambda$. Then there exists a radius R so that the minimization problem

$$\begin{aligned} V=\inf \{{\mathcal {H}}^{3}_{g'}(\partial \Omega ):\Omega \subset {\mathbb {R}}^2\times S^2\text { open, containing }B_1(0)\times S^2\} \end{aligned}$$

(27)

is solved by an open set lying within $B_R(0)\times S^2\subset {\mathbb {R}}^2\times S^2$.

Proof

A solution to problem (27) is known as an outer minimizing hull of the region $B_1(0)\times S^2$. According to a result of Fogagnolo-Mazzieri [14, Theorem 1.1], such an outward minimizing hull exists and is given by a bounded subset of ${\mathbb {R}}^2\times S^2$ so long as $({\mathbb {R}}^2\times S^2,g')$ satisfies the following Euclidean isoperimetric inequality: there is a constant $C>0$ so that

$$\begin{aligned} {\mathcal {H}}^4_{g'}(\Omega )\le C{\mathcal {H}}^{3}_{g'}(\partial \Omega )^{\frac{4}{3}} \end{aligned}$$

(28)

for all bounded regions $\Omega \subset {\mathbb {R}}^2\times S^2$ with smooth boundary. As such, we aim to establish (28).

Leveraging the uniformity assumption (26), it suffices to establish an Euclidean isoperimentric inequality for the product manifold $M_*:=({\mathbb {R}}^2\times S^2,g_{{\mathbb {H}}^2}+g_{S^2})$. To this end, suppose we are given an open set $\Omega \subset M_*$ with smooth boundary. When $\Omega$ has sufficiently small volume, the inequality (28) follows from a classical argument using the fact that $M_*$ satisfies a Ricci curvature lower bound $\textrm{Ric}^{M_*}\ge -g$ and a noncollapsing condition $\textrm{Vol}_{M_*}(B_1(p))\ge 1$. Namely, the result [23, Lemma 3.2] implies there is an $\eta >0$ and constant C so that any $\Omega \subset M_*$ with ${\mathcal {H}}^4(\Omega )\le \eta$ satisfies inequality (28). Now assume ${\mathcal {H}}^4(\Omega )\ge \eta$. The result [15, Theorem 6.19] states that such $\Omega$ satisfy the same isoperimetric inequality as the one satisfied by the fundamental group of any manifold covered by $M_*$. Since $M_*$ covers manifolds with hyperbolic fundamental group, we conclude the existence of a $C'>0$ such that the linear inequality ${\mathcal {H}}^4(\Omega )\le C'{\mathcal {H}}^3(\partial \Omega )$ holds, from which the Euclidean inequality (28) quickly follows. $\square$