Year: 2026
Type: preprint
Abstract: We show that a closed spin four-manifold with nonzero signature, even after being punctured along a finite graph, does not admit a complete metric with uniformly positive scalar curvature. We also show that this implies Schoen's conjecture on $L^{\infty}$-metrics with non-negative scalar curvature for these manifolds. This is in contrast to higher dimensions. A key new observation, which could be of independent interest, is that Gromov's $μ$-bubbles are a convenient class of boundaries for the APS boundary value problem for the Dirac operator.
Source: arXiv (Cornell University)
Authors: Kumar, Aditya, Zhang, Boyu
Institution: University of Maryland, College Park
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214885534