Entropy and Stability of Hyperbolic Manifolds

• Published in: Geometric and functional analysis Vol. 35; no. 3; pp. 877 - 914
• Main Author: Song, Antoine
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.06.2025; Springer Nature B.V
• Subjects: Analysis; Entropy; Manifolds (mathematics); Mathematics; Mathematics and Statistics; Topology; Hyperbolic manifolds; Volume entropy; Metric currents; Plateau problem
• ISSN: 1016-443X; 1420-8970
• DOI: 10.1007/s00039-025-00711-3

Let ( M , g 0 ) 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 g 0 , its volume entropy h ( g ) satisfies h ( g )≥ n −1 with equality only when g is isometric to g 0 . We show that the hyperbolic metric g 0 is stable in the following sense: if g i is a sequence of Riemaniann metrics on M of same volume as g 0 and if h ( g i ) converges to n −1, then there are smooth subsets Z i ⊂ M such that both and tend to 0, and ( M ∖ Z i , g i ) converges to ( M , g 0 ) in the measured Gromov-Hausdorff topology. The proof relies on showing that any spherical Plateau solution for M is intrinsically isomorphic to .