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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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