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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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##### References:
  [Bav-Pan] Bavard, Ch., Pansu, P.: Le volume minimal deR 2. Ann. Sci. Ec. Norm. Super.19, 479-490 (1986) · Zbl 0611.53038  [Be] Besse, A.: Einstein Manifolds, Ergebnisse der Math. und ihrer Grenzgebiete. Band 10. Berlin, Heidelberg, New York: Springer 1987 · Zbl 0613.53001  [Be2] Besse, A.: Manifolds all of whose geodesics are closed, Ergebnisse der Math. und ihrer Grenzgebiete. Band 93. Berlin, Heidelberg, New York: Springer 1978 · Zbl 0387.53010  [Bro] Brooks, R.: The fundamental group and the spectrum of the Laplacian. Comment. Math. Helv.56, 581-598 (1981) · Zbl 0495.58029 · doi:10.1007/BF02566228  [Din] Dinaburg, E.T.: On the relations among various entropy characteristics of dynamical systems. Math. USSR, Izv.5, 337-378 (1971) · Zbl 0248.58007 · doi:10.1070/IM1971v005n02ABEH001050  [E-I] El Soufi, A., Ilias, S.: Operateurs de Shrödinger sur une variété riemannienne et volume conforme Grenoble: Prépublication de l’Institut Fourier no 83 (1987)  [Eb] Ebin, D.: The space of riemannian metrics. Proc. Symp. Am. Math. Soc.15 Global Analysis 11-40 (1968)  [Ga-Me] Gallot, S., Meyer, D.: Operateur de courbure et laplacien des formes différentielles d’une variété riemannienne. J. Math. Pures Appl.54, 259-284 (1975) · Zbl 0316.53036  [Gro1] Gromov, M.: Volume and Bounded cohomology. Publ. Math. Inst. Hautes Étud. Sci.56, 213-307 (1981)  [Gro2] Gromov, M.: Groups of polynomial growth and expanding maps. Publ. Math. Inst. Hautes Étud. Sci.53, 53-78 (1981) · Zbl 0474.20018 · doi:10.1007/BF02698687  [Gro3] Gromov, M.: Filling Riemannian Manifolds. J. Differ. Geom.18, 1-47 (1983) · Zbl 0515.53037  [Gro-Laf-Pan] Gromov, M., Lafontaine, J., Pansu, P.: Structures métriques pour les variétés riemanniennes. Cedic: Fernand Nathan, 1981  [Gui] Guivarc’h, Y.: Sur la représentation intégrale des fonctions harmoniques et des fonctions propres positives dans un espace riemannien symétrique. Bull. Soc. Math. Fr. 2e série,108, 373-392 (1984)  [Ha] Hamenstädt, U.: Metric and topological entropy of geodesic flows. Caltech 1989 (Preprint)  [Hel] Helgason, S.: Differential geometry and symmetric spaces. New York, London: Academic Press 1962 · Zbl 0111.18101  [Ka-Kni-We] Katok, A., Knieper, G., Weiss, H.: Regularity of Topological Entropy (Preprint)  [Kni] Knieper, G.: A second derivative formula of the measure theoretic entropy at spaces of constant negative curvature. Preprint Göttingen  [Ko] Koiso, N.: A decomposition of the space of Riemannian Metrics on a Manifold. Osaka J. Math.16, 423-429 (1979) · Zbl 0416.58007  [Kat] Katok, A.: Four applications of conformal equivalence to Geometry and Dynamics. Ergodic Theory Dyn. Syst.8, 139-152 (1988) · Zbl 0668.58042 · doi:10.1017/S0143385700009391  [Led] Ledrappier, F.: Harmonic measures and Bowen-Margulis measures. Univ. Paris VI (1988) (Preprint)  [Man] Manning, A.: Topological entropy for geodesic flows. Ann. Math.110, 567-573 (1979) · Zbl 0426.58016 · doi:10.2307/1971239  [Ol] Olshanetski, M.A.: Frontière de Martin de l’opérateur de Laplace Beltrami sur les espaces riemanniens symétriques de courbure non positives. Math. Nauk. 1969  [Pan] Pansu, P.: Effondrement des variétés riemanniennes. Séminaire Bourbaki, exposé 618 novembre 1983  [Pes] Pesin, B.: Equations for the entropy of the geodesic flow on a compact riemanian manifold without conjugate points. Math. notes24, 796-805 (1978) · Zbl 0411.58016  [Si] Simons, J.: On the transitivity of holonomy systems. Ann. Math.76, 213-234 (1962) · Zbl 0106.15201 · doi:10.2307/1970273  [Sun] Sunada, T.: Unitary representations of fundamental groups and the spectrum of twisted Laplacians. Topology,28, 2, 1-125 (1989) · Zbl 0681.53025 · doi:10.1016/0040-9383(89)90015-3  [Zim] Zimmer, R.J.: Ergodic theory and semi-simple groups. Boston: Birkhäuser 1984
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