Desingularizing positive scalar curvature 4-manifolds (Q6634453)

A measurable section of positive definite, bilinear forms \(g\) is a uniformly Euclidean metric, in symbols \( g \in L^\infty_E(M)\), if there is a smooth metric \(g_0\) and a positive constant \(C\) such that \(C^{-1}g_0 \leq g \leq Cg_0\). The following conjecture concerning uniformly Euclidean metrics is attributed to Richard Schoen: \N\NConjecture. Let \(M^n\) be a closed manifold that does not admit a positive scalar curvature metric and let \(S \subset M\) be a smooth, closed, embedded submanifold of codimension at least \(3\). If \(g\) is an \(L_E^\infty(M) \cap C^\infty(M\setminus S)\) metric with \(\mathrm{scal}(g) \geq 0\) on \(M\setminus S\), then \(g\) extends smoothly to a Ricci-flat metric on \(M\). \N\NThe author discusses this conjecture in the special case that \(M\) is a \(4\)-dimensional, and that the singular set consists of a collection of disjointly embedded circles and points, from here on reffered to as tolerable singular set.\N\NThe main result of the paper is that uniformly Euclidean psc metrics on \(4\)-manifolds with tolerable singular sets extend to smooth psc metrics once the singularity set was blown up appropriately. More precisely, the author shows:\N\NTheorem. Let \(M^4\) be a closed, oriented \(4\)-manifold and let \(S \subset M\) be a tolerable singular set. Suppose \(g \in L_E^\infty(M) \cap C^\infty(M\setminus S)\) metric with \(\mathrm{scal}(g) > 0\) on \(M \setminus S\). Then there is exists a smooth, closed, oriented psc manifold \((\bar{M},\bar{g})\) with a degree-\(1\) map \(F \colon \bar{M} \rightarrow M\). Moreover, there is a neighbourhood \(U \subset M\), containing and retracting onto \(S\), so that \(F_{\bar{M}\setminus F^{-1}(U)}\) is a conformal diffeomorphism. \N\NThe proof of Theorem 1 is in part analytical and in part topological. On the analytical side, the author needs to adapt the methods of [\textit{C. Li} and \textit{C. Mantoulidis}, Math. Ann. 374, No. 1--2, 99--131 (2019; Zbl 1418.53042)] to \(1\)-dimensional submanifolds. On the topological side, the author requires knowledge about psc bordism groups of \(1\)-dimensional complexes to guarantee the existence of the degree-\(1\) map. The second main result of this article, which is of independent interest, provides the required information.\N\NTheorem. Let \(S\) be a finite \(1\)-dimensional complex. The \(3\)-dimensional oriented psc-bordism group \(\Omega_3^{SO,+}(S)\) is trivial. \N\NIn Section 2, the author summarises the required background material, which includes the notion of enlargeability and why it is an obstruction against the existence of smooth psc metrics, the conformal Laplacian with minimal boundary conditions and why it yields an existence criterion for psc metrics that are of product form near the boundary, the classification of \(3\)-dimensional psc manifolds, and results about ALE spaces.\N\NSection 3 is devoted to a proof of Theorem 2. This proof heavily relies on the fact that it completely known which \(3\)-dimensional manifolds admit a psc metric and that the moduli space of psc metrics on \(3\)-manifolds is path-connected (if it is non-empty) in order to deduce that the list of generators of \(\Omega_3^{SO,+}(S)\) consists of either \(S^3/\Gamma\) or \(S^2 \times S^1\) equipped with their standard psc metrics and a continuous map to \(S\).\N\NThe author claims in this proof that, because the moduli space of psc metrics \(\mathcal{M}^+(M)\) is path connected, one can find a path of psc metrics between a psc metrics \(g_1\) and \(\Phi^\ast g_0\), where \(\Phi\) is some diffeomorhism. Although true, this is not clear a priori because the canonical projection map from the space of psc metrics to the moduli space of those \(R^+(M) \rightarrow M^+(M)\) does not need to be a (Serre-)fibration. However, the path lifting property follows for example from Ebin's slice theorem, see, e.g.,nProposition 4.6 in [\textit{D. Corro} and \textit{J.-B. Kordaß}, Contemp. Math. 775, 65--84 (2021; Zbl 1496.58002)].\N\NSection 4 is devoted to prove Theorem 1. The essential idea here is to perturb the Riemannian metric in a sufficiently small tubular neighbourhood of the singular set such that one can find a separating, stable minimal hypersurface \(\Sigma\) (away from the singular set). The open part that contains \(S\) is called \(A^{\mathrm{in}}\). After a global conformal change of the metric on \(M \setminus A^{\mathrm{in}}\) such that is of product metric near \(\Sigma\), on can replace \(A^{\mathrm{in}}\) by a manifold with boundary \(\Sigma\) such that the metric on \(\Sigma\) extends to a psc metric with product structure near the boundary, and such that the (restriction of) the projection of the tubular neighbourhood \(\Sigma \rightarrow S\) extends to the manifold with boundary, which essentially yields the degree-\(1\) map \(\bar{M} \rightarrow S\).\N\NSection 5 proves some applications of the main results. The first application is that psc metrics with tolerable singular sets cannot exists on enlargeable \(4\)-manifolds. The second application is that the positive mass theorem in dimension \(4\) also holds for these kinds of metrics.\N\NFinally, it should be emphasised once more that this article relies heavily on low-dimensional techniques and one should not expect similar results in higher dimensions. Indeed, in the recent preprint [``Positive scalar curvature with point singularities'', Preprint, \url{arXiv:2407.20163}] \textit{S. Cecchini} et al. gave a counterexample to the above mentioned conjecture in dimension \(8\) and above.