Entropy and stability of hyperbolic manifolds

TL;DR

This paper demonstrates the stability of hyperbolic metrics on closed manifolds of dimension at least 3, showing that metrics with entropy close to the minimal value approximate the hyperbolic metric in a measured Gromov-Hausdorff sense.

Contribution

It proves the stability of hyperbolic metrics under entropy convergence and characterizes spherical Plateau solutions as isomorphic to scaled hyperbolic metrics.

Findings

Hyperbolic metric is stable under entropy convergence.

Sequences of metrics with entropy approaching the minimum approximate the hyperbolic metric.

Spherical Plateau solutions are isomorphic to scaled hyperbolic metrics.

Abstract

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 $h(g)\geq n-1$ 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 $Z_i\subset M$ such that both $\mathrm{Vol}(Z_i,g_i)$ and $\mathrm{Area}(\partial Z_i,g_i)$ tend to $0$, and $(M\setminus Z_i,g_i)$ 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 $(M,\frac{(n-1)^2}{4n} g_0)$.