Local entropy rigidity for hyperbolic manifolds
type: Item
instance of: scholarly article
title: Local entropy rigidity for hyperbolic manifolds
zbMATH Open document ID: 0852.58068
author: Livio Flaminio
published in: Communications in Analysis and Geometry
publication date: 1996-12-09T00:00:00Z
review text: 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 \(g_0\).
Mathematics Subject Classification ID: 37D40
Mathematics Subject Classification ID: 53D25
Mathematics Subject Classification ID: 58E11
Mathematics Subject Classification ID: 54C70
zbMATH DE Number: 870991
zbMATH Keywords: closed hyperbolic manifolds
zbMATH Keywords: topological entropy
zbMATH Keywords: metric entropy
zbMATH Keywords: variations of metrics
MaRDI profile type: Publication
DOI: 10.4310/CAG.1995.V3.N4.A2
description: scientific article; zbMATH DE number 870991