Entropy and Stability of Hyperbolic Manifolds

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