Local entropy rigidity for hyperbolic manifolds
The author considers \(C^5\)-metrics of constant volume on a closed manifold \(M\) which admits a metric \(g_0\) of constant curvature \(-1\). The locally symmetric metric \(g_0 \) is a critical point for the functions \(\text{Ent}_{\text{Liouv}}\) and \(\text{Ent}_{\text{top}}\), which associate to a metric \(g\) the metric entropy resp. the topological entropy of its geodesic flow on the space \(SM\) of directions of \(M\). The author computes the second derivative of the function \(\varphi = \text{Ent}_{\text{top}} - \text{Ent}_{\text{Liouv}}\) and shows that \(\varphi\) is locally strictly convex at \(g_0\) along a \(C^2\)-curve \(g_\varepsilon\) of metrics which is not tangent at \(g_0\) to the orbit of \(g_0\) under the diffeomorphism group. The proof consists of a rather lengthy computation of the second derivative of \(\varphi\) at \(g_0\) in a symbolic representation of the geodesic flow along the lines previously developed by Pollicott, Katok, Contreras and others. Furthermore, using representation theory for the group \(SO(1,3)\) the author derives the existence of a cocompact lattice \(\Gamma\) in \(SO(1,3)\) such that the second derivative of \(\text{Ent}_{\text{Liouv}}\) at the hyperbolic metric \(g_0\) on \(\Gamma \backslash SO (1,3)/SO(3)\) is indefinite. This gives an interesting contrast to the fact that \(\text{Ent}_{\text{top}}\), as a function on metrics of fixed volume, assumes a unique global minimum at \(g_0\).
scientific article; zbMATH DE number 870991