Spin and scalar curvature on punctured four manifolds

We show that a closed spin four-manifold with nonzero signature, even after being punctured, does not admit a complete metric with uniformly positive scalar curvature. In particular, the punctured $K3$ surface does not admit such a metric. This is known to be false in higher dimensions. A key new observation, which could be of independent interest, is that Gromov’s $\mu$-bubbles are a convenient class of boundaries for the APS boundary value problem for the Dirac operator.