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Volume et entropie minimale des espaces localement symétriques. (Minimal volume and entropy of locally symmetric spaces). (French) Zbl 0723.53029
M. Gromov [Publ. Math., Inst. Hautes Etud. Sci. 56, 5-99 (1982; Zbl 0516.53046)] introduced the concept of minimal volume of a Riemannian manifold, \(\min vol(M):=\inf \{Vol(M,g);\;|\) sectional curvature of \(g| \leq 1\}.\) The following questions are immediate: (i) which condition for a differentiable manifold M implies \(\min vol(M)=0\) and which \(\min vol(M)\neq 0?\) (ii) Do we have \(\min vol(M)=Vol(M,g)\) for some suitable metric g with \(|\) sectional curvature of \(g| \leq 1?\) M. Gromov also defined the invariant \(\| M\|:=\inf \{\sum | a_ i|;\;\sum a_ i\sigma_ i\) represents the fundamental class of M with \(a_ i\in {\mathbb{R}}\) and \(\sigma_ i\) being singular simplexes} and proved that for complete Riemannian manifolds, \((n-1)^ nn! \min vol(M)\geq \| M\|.\) He formulated the conjecture: for a manifold admitting a hyperbolic metric, hyp, with finite volume we have \(\min vol(M)=Vol(M,hyp).\)
One of the important aims of this paper is to give an approach to an answer to (ii). The authors consider a compact Riemannian manifold \((M,g_ 0)\) of dimension \(\geq 3\) being Einstein with negative sectional curvature or irreducible locally symmetric with non-positive sectional curvature, i.e. of non-compact type. They prove that there exists a neighbourhood U of \(g_ 0\) in the space of all Riemannian metrics on M such that:
A) \(K(g)\geq K(g_ 0)\) for \(g\in U,\) \(K(g)=\int_{M}| scal g|^{n/2}dv_ g,\) and the equality holds iff g is isometric to \(g_ 0\).
B) There exists a constant \(C(g_ 0)\) with \((d(g,g_ 0))^ 2\leq C(g_ 0)| K(g)-K(g_ 0)|\) for \(g\in U\), \(d(g,g_ 0)=\inf \{\| \phi^*(g)-g_ 0\|; \phi \in Diff(M).\)
C) \(Vol(g)\geq Vol(g_ 0)\) when \(g\in U\) and \(scal(g)\geq scal(g_ 0).\) D) If M is compact, then \(n!(n-1)^ n\min vol(M)\geq n^{n/2}\| M\|.\)
Following Gromov the authors consider also the invariants \(h_ V(M,g):=\lim_{R\to \infty}((1/R)\log Vol(\tilde B(x,R))),\) where \(\tilde B(x,R)\) is the ball with center x and radius R in the universal Riemannian covering \((\tilde M,\tilde g)\) of \((M,g)\), and \(Min ent(M):=\inf \{h_ V(M,g);\;Vol(M,g)=1\}\) called minimal entropy. By means of an equivariant immersion of \((\tilde M,\tilde g)\) into the unit sphere of a Hilbert space the authors introduce a new topological invariant called the spherical volume of M and use it to prove the following conjecture due to Gromov: if M admits a locally symmetric metric \(g_ 0\) of non-compact type, then \(Min ent(M)=h_ V(M,g_ 0).\) The paper contains many interesting relationships among the minimal entropy, the minimal volume and the spherical volume as well.

MSC:
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
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