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Dirac eigenvalues and the hyper-hemispherical radius

  • Pengshuai Shi
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Abstract

On a connected compact Riemannian spin manifold with boundary, we prove an upper estimate for the smallest eigenvalue of the Dirac operator under the boundary condition associated with a chirality operator. This upper bound is expressed in terms of the hyper-hemispherical radius. Moreover, the upper bound is realized if and only if the manifold is a hemisphere. As an application, we get a scalar curvature rigidity result for the hemisphere in the case of Lipschitz maps whose differential has a prescribed trace-norm bound. Our estimate extends Bär's result for closed manifolds, where the upper bound is given in terms of hyperspherical radius.

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Cite

BibTeXRIS
BibTeX
@misc{shi2026dirac,
  title = {{Dirac eigenvalues and the hyper-hemispherical radius}},
  author = {Pengshuai Shi},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.32117},
  eprint = {2610.32117},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.32117},
}
RIS
TY  - UNPB
TI  - Dirac eigenvalues and the hyper-hemispherical radius
AU  - Pengshuai Shi
PY  - 2026
DA  - 2026/10/23
PB  - arXiv
DO  - 10.48550/arxiv.2610.32117
UR  - https://arxiv.org/abs/2610.32117
AB  - On a connected compact Riemannian spin manifold with boundary, we prove an upper estimate for the smallest eigenvalue of the Dirac operator under the boundary condition associated with a chirality operator. This upper bound is expressed in terms of the hyper-hemispherical radius. Moreover, the upper bound is realized if and only if the manifold is a hemisphere. As an application, we get a scalar curvature rigidity result for the hemisphere in the case of Lipschitz maps whose differential has a prescribed trace-norm bound. Our estimate extends Bär's result for closed manifolds, where the upper bound is given in terms of hyperspherical radius.
ER  -