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Torus and Positive Mass stability for metrics with Ricci curvature lower bound

  • Edward Bryden
  • Zhizhang Xie
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Abstract

Consider a sequence of metrics gi on the torus whose members have uniform lower bounds on their first stable systoles and Ricci curvatures, and have a uniform upper bound on their diameters. If the L1 norm of the negative part of the scalar curvatures vanishes along this sequence, then we show there is a subsequence of the metrics which converges in the measured-Gromov-Hausdorff topology to a flat metric on the torus. Something analogous holds for sequences of asymptotically flat spin Riemannian manifolds with a uniform lower bound on Ricci curvature and nonnegative scalar curvature: if the ADM masses of the distinguished ends tend to zero along the sequence, then the manifolds converge to Euclidean space in the pointed measured Gromov-Hausdorff sense.

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Cite

BibTeXRIS
BibTeX
@misc{bryden2026torus,
  title = {{Torus and Positive Mass stability for metrics with Ricci curvature lower bound}},
  author = {Edward Bryden and Zhizhang Xie},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.09067},
  eprint = {2608.09067},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.09067},
}
RIS
TY  - UNPB
TI  - Torus and Positive Mass stability for metrics with Ricci curvature lower bound
AU  - Edward Bryden
AU  - Zhizhang Xie
PY  - 2026
DA  - 2026/09/02
PB  - arXiv
DO  - 10.48550/arxiv.2608.09067
UR  - https://arxiv.org/abs/2608.09067
AB  - Consider a sequence of metrics g_i on the torus whose members have uniform lower bounds on their first stable systoles and Ricci curvatures, and have a uniform upper bound on their diameters. If the L^1 norm of the negative part of the scalar curvatures vanishes along this sequence, then we show there is a subsequence of the metrics which converges in the measured-Gromov-Hausdorff topology to a flat metric on the torus. Something analogous holds for sequences of asymptotically flat spin Riemannian manifolds with a uniform lower bound on Ricci curvature and nonnegative scalar curvature: if the ADM masses of the distinguished ends tend to zero along the sequence, then the manifolds converge to Euclidean space in the pointed measured Gromov-Hausdorff sense.
ER  -