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PreprintarXiv licencearXiv

Perelman's entropy and Gromov-Hausdorff versus dynamical stability of round spheres

  • Reto Buzano
  • Shengwen Wang
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Abstract

We study the Gromov-Hausdorff stability of round spheres under perturbations of Perelman's nu-entropy. We show that in dimension two almost maximal entropy, together with an area normalisation, implies Gromov-Hausdorff closeness to the round two-sphere. In contrast, in dimensions at least three, we construct smooth metrics on the sphere whose entropies converge to that of the round sphere, but which remain a definite Gromov-Hausdorff distance away from it. These metrics can be chosen conformal and uniformly bi-Lipschitz equivalent to the round metric, with the same volume as the round sphere. Finally, in dimension three, we contrast this Gromov-Hausdorff instability with a dynamical stability result: metrics with nu-entropy above that of the round shrinking cylinder converge exponentially fast to round metrics under the volume-normalised Ricci flow.

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Cite

BibTeXRIS
BibTeX
@misc{buzano2026perelmans,
  title = {{Perelman's entropy and Gromov-Hausdorff versus dynamical stability of round spheres}},
  author = {Reto Buzano and Shengwen Wang},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.06678},
  eprint = {2611.06678},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.06678},
}
RIS
TY  - UNPB
TI  - Perelman's entropy and Gromov-Hausdorff versus dynamical stability of round spheres
AU  - Reto Buzano
AU  - Shengwen Wang
PY  - 2026
DA  - 2026/11/04
PB  - arXiv
DO  - 10.48550/arxiv.2611.06678
UR  - https://arxiv.org/abs/2611.06678
AB  - We study the Gromov-Hausdorff stability of round spheres under perturbations of Perelman's nu-entropy. We show that in dimension two almost maximal entropy, together with an area normalisation, implies Gromov-Hausdorff closeness to the round two-sphere. In contrast, in dimensions at least three, we construct smooth metrics on the sphere whose entropies converge to that of the round sphere, but which remain a definite Gromov-Hausdorff distance away from it. These metrics can be chosen conformal and uniformly bi-Lipschitz equivalent to the round metric, with the same volume as the round sphere. Finally, in dimension three, we contrast this Gromov-Hausdorff instability with a dynamical stability result: metrics with nu-entropy above that of the round shrinking cylinder converge exponentially fast to round metrics under the volume-normalised Ricci flow.
ER  -