Almost maximal volume entropy (Q1749383)

This article gives examples to show that rigidity theorems of \textit{F. Ledrappier} and \textit{X. Wang} [J. Differ. Geom. 85, No. 3, 461--477 (2010; Zbl 1222.53040)] and \textit{L. Chen} et al. [``Quantitative Volume space form rigidity under lower Ricci curvature bound'', Preprint, \url{arXiv:1604.06986}] for compact, \(n\)-dimensional Riemannian manifolds with \(\operatorname{Ric} \geq -(n-1)\) are false if upper bounds for the diameter in those results are replaced by upper bounds for the volume. Given a compact Riemannian manifold \(M\) one can define the \textit{volume entropy} \[ h_{v}(M) = \lim_{r \rightarrow \infty} \frac{1}{r}\log \operatorname{vol}(B_{r}(\tilde{x})), \] where \(B_{r}(\tilde{x})\) is the ball of radius \(r\) centered at a point \(\tilde{x}\) in the universal Riemannian cover \(\tilde{M}\) of \(M\). \textit{A. Manning} [Ann. Math. (2) 110, 567--573 (1979; Zbl 0426.58016)] showed that this limit always exists and is independent of the point \(\tilde{x}\) chosen. Ledrappier and Wang [loc. cit.] proved the following. {Theorem 1.} Let \(M\) be a compact \(n\)-dimensional Riemannian manifold with \(\operatorname{Ric} \geq -(n-1)\). Then \(h_{v}(M) \leq (n-1)\) with equality if and only if \(M\) is hyperbolic; i.e., the sectional curvature \(K\) in \(M\) is identically \(-1\). Chen et al. [loc. cit.] proved the following. {Theorem 2.} Given \(n\) and \(d >0\) there exists \(\epsilon(n,d) > 0\) such that if \(\epsilon\) satisfies \(0 \leq \epsilon \leq \epsilon(n,d)\), and if \(M\) is any closed \(n\)-dimensional Riemannian manifold with \(\operatorname{Ric} \geq -(n-1)\), \(\operatorname{diam}(M) \leq d\) and \(h_{v}(M) \geq (n-1) - \epsilon\), then \(M\) is diffeomorphic to a hyperbolic manifold. In this article the authors prove the following. {Theorem 3.} For every \(n \geq 3\) there exist \(v > 0\) and a sequence \(\{M_{i} \}\) of closed Riemannian manifolds that satisfy the following conditions: {\parindent=6mm \begin{itemize}\item[(1)] \(-1 \leq K \leq 0\). \item[(2)] \(\operatorname{vol}(M_{i}) \leq v\) for all \(i\). \item[(3)] \(h_{v}(M_{i}) \geq - (n-1) - \epsilon_{i}\) , where \(\epsilon_{i} \rightarrow 0\) as \(i \rightarrow \infty\). \item[(4)] \(M_{i}\) cannot carry a Riemannian metric with \(K < 0\). \end{itemize}} Note that the first condition implies that \(\operatorname{Ric} \geq -(n-1)\) and the fourth condition implies that \(M_{i}\) cannot be diffeomorphic to a hyperbolic manifold for any \(i\). Theorem 3 is actually a consequence of the authors' result: {Theorem 4.} For every \(n \geq 3\) there exists \(v > 0\) such that for all \(\epsilon > 0\) there exists a compact \(n\)-dimensional \(M = M_{\epsilon}\) with the following properties: {\parindent=6mm \begin{itemize}\item[(1)] \(-1 - \epsilon \leq K \leq 0\) \item[(2)] \(\operatorname{vol}(M) \leq v\). \item[(3)] There exists an open subset \(W\) of \(M\) with \(\operatorname{vol}(W) \geq (1 - \epsilon) \operatorname{vol}(M)\) such that \(K \equiv -1\) on \(W\). \item[(4)] The fundamental group \(\pi_{1}(M)\) has an abelian subgroup of rank \(\geq 2\). \end{itemize}} Condition (4) of Theorem 4 implies, by Preissmann's theorem, Condition (4) of Theorem 3.