Entropy and Stability of Hyperbolic Manifolds
Abstract
Let (M,g0) be a closed oriented hyperbolic manifold of dimension at least 3. By the volume entropy inequality of G. Besson, G. Courtois and S. Gallot, for any Riemannian metric g on M with same volume as g0, its volume entropy h(g) satisfies h(g)≥n−1 with equality only when g is isometric to g0. We show that the hyperbolic metric g0 is stable in the following sense: if gi is a sequence of Riemaniann metrics on M of same volume as g0 and if h(gi) converges to n−1, then there are smooth subsets Zi⊂M such that both Vol(Zi,gi) and Area(∂Zi,gi) tend to 0, and (M∖Zi,gi) converges to (M,g0) in the measured Gromov-Hausdorff topology. The proof relies on showing that any spherical Plateau solution for M is intrinsically isomorphic to (M,(n−1)²/4n g₀).
Copyright and License
© The Author(s) 2025.
This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.
Acknowledgement
I am grateful to Gérard Besson, Gilles Courtois, Juan Souto, John Lott, Ursula Hamenstädt, Ben Lowe and Demetre Kazaras for insightful discussions during the writing of this article. I would especially like to thank Cosmin Manea, Hyun Chul Jang, Xingzhe Li and Dongming (Merrick) Hua for their careful reading, suggestions and for several corrections.
Funding
A.S. was partially supported by NSF grant DMS-2104254. This research was conducted during the period A.S. served as a Clay Research Fellow.
Files
s00039-025-00711-3.pdf
Files
(1.8 MB)
| Name | Size | Download all |
|---|---|---|
|
md5:e60eaf88e21f6b85ce89801af372b48b
|
1.8 MB | Preview Download |
Additional details
Related works
- Describes
- Journal Issue: https://rdcu.be/ennlk (URL)
- Is new version of
- Discussion Paper: arXiv:2302.07422 (arXiv)
Funding
- National Science Foundation
- DMS-2104254
- Clay Mathematics Institute
Dates
- Accepted
-
2025-04-14
- Available
-
2025-05-14Published