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 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 \(\operatorname{Vol}(Z_{i},g_{i})\) and \(\operatorname{Area}(\partial Z_{i},g_{i})\) 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 \((M,\frac{(n-1)^{2}}{4n} g_{0})\).