PreprintCC BY-NC-ND 4.0arXiv
Spin and scalar curvature on punctured four manifolds
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∞-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.
Related
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BibTeX
@misc{kumar2026spin,
title = {{Spin and scalar curvature on punctured four manifolds}},
author = {Aditya Kumar and Boyu Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.37948},
eprint = {2609.37948},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.37948},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Spin and scalar curvature on punctured four manifolds
AU - Aditya Kumar
AU - Boyu Zhang
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2609.37948
UR - https://arxiv.org/abs/2609.37948
AB - 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^∞-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.
ER -