Minimal entropy and Mostow's rigidity theorems

Abstract

Let (Y, g) be a compact connected n-dimensional Riemannian manifold and let () be its universal cover endowed with the pulled-back metric. If y ∈ , we definewhere B(y, R) denotes the ball of radius R around y in . It is a well known fact that this limit exists and does not depend on y ([Man]). The invariant h(g) is called the volume entropy of the metric g but, for the sake of simplicity, we shall use the term entropy. The idea of recognizing special metrics in terms of this invariant looks at first glance very optimistic. First the entropy, which behaves like the inverse of a distance, is sensitive to changes of scale which makes it a bad invariant: however, this is a minor drawback that can be circumvented by looking at the behaviour of the entropy functional on the space of metrics with fixed volume (equal to one for example). Nevertheless, it seems very unlikely that two numbers, the entropy and the volume, might characterize any metric. The very first person to consider such a possibility was Katok ([Kat1]). In this article the entropy is thought of as a dynamical invariant which actually is suggested by its name. More precisely, let us define this dynamical invariant, which is called the topological entropy: let (M, d) be a compact metric space and ψt, a flow on it, we define.

Perelman’s $${\bar{\lambda}}$$ -invariant and collapsing via geometric characteristic splittings

“…The latter case was shown by Besson, Courtois and Gallot in a series of papers which found deep connections and consequences in the theory, the interested reader is invited to consult [3,4].…”

Asymptotic Conditions for Smooth Rigidity of Negatively Curved Manifolds

“…Under these assumptions on M , the measures σ s x limit to the push forward measures (∂f ) * ν x as s → h(g), where ν x also denotes the Patterson-Sullivan measure on ∂ ∞ M (see [BCG96]). The two conditions simplify to…”

“…Our approach is to exploit and analyze the barycenter construction developed in this generality by Besson Courtois and Gallot (see e.g. [BCG95,BCG96,BCG99]). As a consequence, the resulting homotopy is fairly explicit.…”

Volume entropy and integral Ricci curvatures over closed geodesics

“…In this paper, we obtain an upper bound of the volume entropy and the simplicial volume with integrals of Ricci curvature over closed geodesics instead of a pointwise curvature bound or an integral curvature on the whole space and then apply it to the real Schwarz lemma by Besson,Courtois and Gallot in [BCG1], [BCG2], [BCG3]. The volume entropy h(M ) of M is defined as follows:…”