Entropy and stability of hyperbolic manifolds

2023/02/15 by Song, Antoine · 1 citation
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2302.07422

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 g_i is a sequence of Riemaniann metrics on M of same volume as g₀ and if h(g_i) converges to n-1, then there are smooth subsets Z_i⊂ M such that both Vol(Z_i,g_i) and Area(∂ Z_i,g_i) tend to 0, and (M∖ Z_i,g_i) 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).

Cited by

• Volume Stability for Hyperbolic Manifolds and Applications to General Relativity

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