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Nonexistence of complete metrics with uniformly positive scalar curvature

  • Tsz-Kiu Aaron Chow
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Abstract

Let X be a connected oriented even-dimensional manifold whose universal cover is spin. We prove that X admits no complete metric with uniformly positive scalar curvature when either infinite relative K-area or a relative cohomological condition holds along compact sets escaping to infinity in a fixed open subset with compact complement. The proof compares twisted Dirac operators on spin covers of compact manifolds with boundary and combines a heat-kernel argument on fundamental domains with a long-neck estimate.

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Cite

BibTeXRIS
BibTeX
@misc{chow2026nonexistence,
  title = {{Nonexistence of complete metrics with uniformly positive scalar curvature}},
  author = {Tsz-Kiu Aaron Chow},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.10442},
  eprint = {2609.10442},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.10442},
}
RIS
TY  - UNPB
TI  - Nonexistence of complete metrics with uniformly positive scalar curvature
AU  - Tsz-Kiu Aaron Chow
PY  - 2026
DA  - 2026/09/09
PB  - arXiv
DO  - 10.48550/arxiv.2609.10442
UR  - https://arxiv.org/abs/2609.10442
AB  - Let X be a connected oriented even-dimensional manifold whose universal cover is spin. We prove that X admits no complete metric with uniformly positive scalar curvature when either infinite relative K-area or a relative cohomological condition holds along compact sets escaping to infinity in a fixed open subset with compact complement. The proof compares twisted Dirac operators on spin covers of compact manifolds with boundary and combines a heat-kernel argument on fundamental domains with a long-neck estimate.
ER  -