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arXiv · 2611.06678

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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Reto Buzano, Shengwen Wang. 2026-11-04. Perelman's entropy and Gromov-Hausdorff versus dynamical stability of round spheres. https://arxiv.org/abs/2611.06678

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