## Entropy and rigidity of locally symmetric spaces of strictly negative curvature. (Entropies et rigidités des espaces localement symétriques de courbure strictement négative.)(French)Zbl 0851.53032

In this article the authors prove a major new result whose corollaries settle several difficult outstanding problems or provide new proofs of other significant known results. Theorem. Let $$Y$$ and $$X$$ denote two compact oriented manifolds of the same dimension $$n$$, and let $$f : Y \to X$$ be a continuous map of nonzero degree. Suppose that $$X$$ admits a locally symmetric Riemannian metric $$g_0$$ with strictly negative sectional curvature, and let $$g$$ be any Riemannian metric on $$Y$$. Let $$h(Y,g)$$ and $$h(X,g_0)$$ denote the volume entropy of $$(Y,g)$$ and $$(X,g_0)$$, respectively. Then $$h^n(Y,g) \text{Vol} (Y,g) \geq |\text{deg } f|h^n(X,g_0) \text{Vol} (X,g_0)$$, and equality holds if and only if there exists a positive constant $$c$$ and a Riemannian covering $$f^* : (Y,cg) \to (X,g_0)$$ that is homotopic to $$f$$.
Note that in a Riemannian manifold $$(Z,g)$$ the constant $$h^n(Z,g) \text{Vol}(Z,g)$$ is unchanged if one multiplies the metric $$g$$ by any positive constant.
As easy consequences of the result above the authors obtain several important corollaries. Corollary 1. Let $$Y$$ and $$X$$ denote two compact oriented manifolds of the same dimension $$n$$, and let $$f : Y \to X$$ be a continuous map of nonzero degree. Let $$g_0$$ be a metric on $$X$$ with sectional curvature $$\equiv -1$$, and let $$g$$ be a metric with $$\text{Ricci}(g) \geq -(n -1)g$$. Then $$\text{Vol} (Y,g) \geq |\text{deg }f |\text{Vol}(X,g_0)$$. If equality holds and $$n \geq 3$$, then $$(Y,g)$$ has constant negative sectional curvature and there exists a positive constant $$c$$ and a Riemannian covering $$f^* : (Y,cg) \to (X,g_0)$$ that is homotopic to $$f$$. Corollary 2 (Mostow rigidity theorem). Let $$(Y,g)$$ and $$(X,g_0)$$ be homotopically equivalent, compact oriented Riemannian manifolds with negative sectional curvature. Then $$(Y,cg)$$ and $$(X,g_0)$$ are isometric for some positive constant $$c$$. Corollary 3 (Corlette, Siu, Thurston). Let $$(Y,g)$$ and $$(X,g_0)$$ be compact, negatively curved, rank 1 locally symmetric spaces with the same type (real, complex, quaternionic or Cayley) and with the same upper bound for the sectional curvature. Let $$f : Y \to X$$ be a continuous map of nonzero degree. If $$\text{Vol}(Y,g) = |\text{deg }f|\text{Vol } (X,g_0)$$, then $$f$$ is homotopic to a Riemannian covering $$f^* : (Y,g) \to (X,g_0)$$.
The geodesic flows $$\{g^t_Y\}$$ and $$\{g^t_X\}$$ on the unit tangent bundles $$SY$$ and $$SX$$ of compact Riemannian manifolds $$(Y,g)$$ and $$(X,g_0)$$ are said to be $$C^k$$ conjugate if there exists a $$C^k$$ diffeomorphism $$F : SY \to SX$$ such that $$F \circ g^t_Y = g^t_X \circ F$$ for all $$t\in \mathbb{R}$$. Corollary 4. Let $$(Y,g)$$ and $$(X,g_0)$$ be compact Riemannian manifolds with $$C^1$$ conjugate geodesic flows. If $$(X,g_0)$$ is locally symmetric with negative sectional curvature, then $$(Y,g)$$ is isometric to $$(X,g_0)$$.
A Riemannian manifold $$X$$ is said to be harmonic if all geodesic spheres in $$X$$ have constant mean curvature. If $$X$$ has nonpositive sectional curvature, then $$X$$ is said to be asymptotically harmonic if the horospheres in the universal cover $$\widetilde X$$ have constant mean curvature. Harmonic implies asymptotically harmonic for nonpositively curved manifolds, and in general locally symmetric manifolds are harmonic. Corollary 5. Let $$(Y,g)$$ be compact and asymptotically harmonic with negative sectional curvature. Then $$(Y,g)$$ is a locally symmetric space. Corollary 6. Let $$X$$ be a compact Riemannian manifold with negative sectional curvature whose stable or unstable Anosov foliations in $$SX$$ are $$C^\infty$$. Then $$X$$ is locally symmetric. Corollary 7. Let $$X$$ be a compact Riemannian manifold with negative sectional curvature such that the Bowen-Margulis and harmonic measures are proportional in $$SX$$. Then $$X$$ is locally symmetric.
Corollary 6 brings to completion work of Kanai, who proved the first version of this result. It follows from Corollary 4 and a result of Benoist-Foulon-Labourie. Corollary 7 follows from Corollary 5 and a result of Ledrappier that $$X$$ is asymptotically harmonic under the hypotheses.
For the convenience of the reader we outline a proof of the inequality of the main theorem in the case that $$Y = X$$ and $$f : Y \to X$$ is the identity. The argument in the general case is a modification of this one. Let $$g_0$$ denote the locally symmetric metric of negative sectional curvature on $$X$$ and also on the universal Riemannian cover $$\widetilde X$$. Let $$\partial \widetilde X$$ denote the boundary at infinity that consists of equivalence classes of asymptotic unit speed geodesics relative to $$g_0$$. Fix a point $$p_0$$ in $$\widetilde X$$ and identify $$\partial \widetilde X$$ homeomorphically with $$S_{p_0} \widetilde X$$, the unit vectors of $$\widetilde X$$ at $$p_0$$. Let $$d\theta$$ denote the probability measure on $$\partial \widetilde X$$ that is defined via this identification by the normalized Riemannian measure on $$S_{p_0} \widetilde X$$. Each point $$\theta \in \partial \widetilde X$$ defines a Busemann function $$B_\theta : \widetilde X \to \mathbb{R}$$ given by $$B_\theta (x) = \lim_{t \to \infty} d(x,\gamma_\theta (t)) - t$$, where $$\gamma_\theta$$ is the unit speed representative of $$\theta$$ that starts at $$p_0$$. Each Busemann function $$B_\theta$$ is $$C^2$$ and convex; that is, the restriction of $$B_\theta$$ to each maximal geodesic of $$\widetilde{X}$$ is a convex function on $$\mathbb{R}$$. Define $$B : \widetilde{X} \times \partial \widetilde{X} \to \mathbb{R}$$ by $$B(x,\theta) = B_\theta(x)$$.
Measures on $$\partial \widetilde X$$ and the barycenter map: Let $$\mathcal M$$ denote the collection of nonatomic probability measures on $$\partial \widetilde{X}$$. For each $$\mu \in {\mathcal M}$$ the function $$B_\mu (x) = \int_{\partial \widetilde{X}} B(x,\theta)d\mu$$ is a strictly convex function on $$\widetilde {X}$$ that assumes a minimum at a unique point of $$\widetilde X$$ that we denote by $$\text{Bar}(\mu)$$, the barycenter of $$\mu$$. If $$\Gamma \subseteq I(\widetilde X)$$ is the deckgroup of $$X = \widetilde X/\Gamma$$, then $$\text{Bar} (\gamma^* \mu) = \gamma(\text{Bar} (\mu))$$ for all $$\gamma \in \Gamma$$ and all $$\mu \in {\mathcal M}$$. Let $$S^\infty_+$$ denote the set of positive functions of unit norm in $$L^2 (\partial \widetilde{X}, d \theta)$$, and define $$\pi : S^\infty_+ \to \widetilde{X}$$ by $$\pi(\varphi) = \text{Bar}(\varphi^2 d\theta)$$. The authors show that $$\pi$$ is a $$\Gamma$$-equivariant $$C^1$$ submersion of $$S^\infty_+$$ onto $$\widetilde X$$. In particular, if $$\omega$$ is a closed $$\Gamma$$-equivariant $$n$$-form in $$\widetilde X$$, where $$n = \dim \widetilde X$$, then $$\pi^*(\omega)$$ is a closed, $$\Gamma$$-equivariant $$n$$-form in $$S^\infty_+$$.
Spherical volume and calibrations: Let $$\mathcal N$$ consist of the $$\Gamma$$-equivariant Lipschitz maps $$\Phi$$ from $$\widetilde{X}$$ to $$S^\infty_+$$. The differential $$(d\Phi)_x$$ exists for almost all $$x$$ in $$\widetilde X$$. If $$\alpha$$ is an $$n$$-form on $$S^\infty_+$$, then define
(1) $$\text{comass}(\alpha) = \sup\{|a_f(U_1,\dots, U_n)|: f \in S^\infty_+$$ and $$\{U_1,\dots, U_n\}$$ are orthonormal vectors in $$T_f S^\infty_+\}$$.
If $$\alpha$$ is a $$\Gamma$$-equivariant $$n$$-form on $$S^\infty_+$$ and $$\Phi$$ is an element of $$\mathcal N$$, then the $$\Gamma$$-equivariant pullback $$\Phi^*(\alpha)$$ is defined almost everywhere in $$\widetilde{X}$$ and induces an $$n$$-form in $$X$$ that by abuse of notation we also denote by $$\Phi^*(\alpha)$$. We define $$V_\Phi(\alpha) = (1/ \text{comass}(\alpha)) \int_X \Phi^* (\alpha)$$. Next we define
(2) $$\text{Vol} (\Phi) = \sup\{V_\Phi (\alpha) : \alpha$$ is a $$\Gamma$$-equivariant $$n$$-form in $$S^\infty_+\}$$, where $$\Phi \in {\mathcal N}$$; $$\text{Sphere Vol} (X) = \inf\{\text{Vol}(\Phi) : \Phi \in {\mathcal N}\}$$.
The authors prove
(3) $$\text{Sphere Vol} (X) \leq (h^2(g)/4n)^{n/2} \text{Vol}(X,g)$$ for every metric $$g$$ on $$X$$, where $$h(g)$$ denotes the volume entropy of $$g$$.
If $$\omega^*$$ is any $$n$$-form on $$S^\infty_+$$ and $$\Phi$$ is any element of $$\mathcal N$$, then by (1) $$\Phi^* (\omega^*)_x (e_1,\dots, e_n) \leq \text{comass} (\omega^*)\cdot |d\Phi^* (e_1) \wedge \cdots \wedge d\Phi^*(e_n)|$$ for all points $$x$$ in $$\widetilde{X}$$ and all orthonormal bases $$\{e_1,\dots, e_n\}$$ in $$T_x \widetilde X$$. If $$\omega^*$$ is closed, then $$\omega^*$$ is said to calibrate a map $$\Phi^* \in {\mathcal N}$$ if equality holds above for all $$x$$ and $$\{e_1,\dots, e_n\}$$. From (1) and (2) one obtains
(4) (Proposition 4.3) Let $$\omega^*$$ be a closed $$\Gamma$$-equivariant $$n$$-form on $$S^\infty_+$$ that calibrates an element $$\Phi^*$$ of $$\mathcal N$$. Then $$\text{Vol} (\Phi) \geq \text{Vol}(\Phi^*) = \text{Sphere Vol} (X)$$ for all $$\Phi \in {\mathcal N}$$.
Now define $$\Phi_0 \in {\mathcal N}$$ by $$\Phi_0(x) = (p_0(x,\cdot))^{1/2}$$, where $$p_0 : \widetilde X \times \partial \widetilde{X} \to \mathbb{R}$$ denotes the Poisson kernel given by $$p_0(x,\theta) = \text{exp}(-h(g_0) \cdot B(x,\theta))$$, where $$h(g_0)$$ is the volume entropy of $$g_0$$. In Proposition 5.7 the authors prove
(5) If $$\omega_0$$ is the closed, $$\Gamma$$-equivariant volume $$n$$-form on $$\widetilde{X}$$ determined by $$g_0$$, then $$\pi^*(\omega_0)$$ is a closed, $$\Gamma$$-equivariant $$n$$-form on $$S^\infty_+$$ that calibrates $$\Phi_0$$, where $$\pi : S^\infty_+ \to \widetilde {X}$$ is defined above.
Finally, from (3), (4), (5) and the calculations in example 2.6a one obtains $$h(g)^n \text{Vol}(X,g) \geq (4n)^{n/2} \text{Sphere Vol}(X) = (4n)^{n/2} \text{Vol}(\Phi_0) = h(g_0)^n \text{ Vol}(X,g_0).$$

### MSC:

 53C35 Differential geometry of symmetric spaces
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### References:

 [1] [B]Y. Babenko, Closed geodesics, asymptotic volume, and characteristics of group growth, Math. USSR Izvestya 33 (1989), 1–37. · Zbl 0668.53032 [2] [BaGrS]W. Ballmann, M. Gromov, V. Schroeder, Manifolds of Nonpositive Curvature, Birkhäuser Boston Inc. 1985. · Zbl 0591.53001 [3] [BePe]R. Benedetti, C. Petronio, Lectures on Hyperbolic Geometry, Universitext, Springer Verlag 1993. [4] [BenFoL]Y. Benoist, P. Foulon, F. Labourie, Flots d’Anosov à distributions stable et instable différentiables, J. Amer. Math. Soc. 5:1 (1992), 33–74. · Zbl 0759.58035 [5] [Ber]M. Berger, Du côté de chez Pu, Ann. Sci. École Norm. Sup. 5 (1972), 1–44. · Zbl 0227.52005 [6] [BerGaMaz]M. Berger, P. Gauduchon, E. Mazet, Le spectre d’une variété riemannienne, Springer Lecture Notes in Mathematics 194, (1971). · Zbl 0223.53034 [7] [Bes1]A.L. Besse, Manifolds All of Whose Geodesics are Closed, Ergebnisse der Math., Springer-Verlag, 1978. · Zbl 0387.53010 [8] [Bes2]A.L. Besse, Einstein Manifolds, Ergebnisse der Math., Springer-Verlag, 1987. [9] [BoW]M. Boileau, S. Wang, Non-zero degree maps and surface bundles overS 1, preprint, Université Paul Sabatier, Toulouse, 1995. [10] [BsCoG1]G. Besson, G. Courtois, S. Gallot, Le volume et l’entropie minimal des espaces localement symétriques, Invent. Math. 103 (1991), 417–445. · Zbl 0723.53029 [11] [BsCoG2]G. Besson, G. Courtois, S. Gallot, Les variétés hyperboliques sont des minima locaux de l’entropie topologique, Invent. Math. 117 (1994), 403–445. · Zbl 0814.53031 [12] [BsCoG3]G. Besson, G. Courtois, S. Gallot, Sur le volume des simplexes hyperboliques idéaux, en préparation. [13] [BuK]K. Burns, A. Katok, Manifolds with non-positive curvature, Ergod. Th. of Dynam. Sys. 5 (1985), 307–317. · Zbl 0572.58019 [14] [C]K. Corlette, Archimedean superrigidity and hyperbolic geometry, Ann. of Math. 135 (1992), 165–182. · Zbl 0768.53025 [15] [CrKl]C. Croke, B. Kleiner, Conjugacy and rigidity for manifolds with a parallel vector field, preprint, février 1992. [16] [Cr]C. Croke, Rigidity for surfaces of non-positive curvature, Comm. Math. Helv. 65:1 (1990), 150–169. · Zbl 0704.53035 [17] [CdV]Y. Colin de Verdière, Spectre du laplacien et longueurs des géodésiques périodiques, Compositio Math. 27 (1973), 159–184. [18] [DRi]E. Damek, F. Ricci, A class of non-symmetric harmonic riemannian spaces, Bull. Amer. Math. Soc. 27 (1992), 139–142. · Zbl 0755.53032 [19] [DoE]E. Douady, C. Earle, Conformally natural extension of homeomorphisms of the circle, Acta. Math. 157 (1986), 23–48. · Zbl 0615.30005 [20] [F]H. Federer, Geometric Measure Theory, Grundlehren Band, Springer Verlag, 153 (1969). · Zbl 0176.00801 [21] [Fo]P. Foulon, Nouveraux invariants géométriques des systèmes dynamiques du second ordre: application à l’étude du comportement ergodique, Thèse d’état, École Polytechnique, Centre de Mathématiques (1986). [22] [FoL]P. Foulon, F. Labourie, Sur les variétés compactes asymptotiquement harmoniques, Invent. Math. 109 (1992), 97–111. · Zbl 0767.53030 [23] [Fu]H. Furstenberg, A Poisson formula for semi-simple Lie groups, Annals of Math. 77 (1963), 335–386. · Zbl 0192.12704 [24] [GHuLa]S. Gallot, D. Hulin, J. Lafontaine, Riemannian Geometry, Universitext, Springer-Verlag (1987). [25] [Gr1]M. Gromov, Volume and bounded cohomology, Publ. Math. Inst. Hautes Étud. Sci. 56 (1981), 213–307. [26] [Gr2]M. Gromov, Filling riemannian manifolds, J. Differ. Geom. 18 (1983), 1–147. · Zbl 0515.53037 [27] [GrT]M. Gromov, W. Thurston, Pinching constants for hyperbolic manifolds, Invent. Math. 89 (1987), 1–12. · Zbl 0646.53037 [28] [H1]U. Hamenstädt, Time-preserving conjugacies of geodesic flows, Ergod. Th. & Dynam. Sys. 12 (1992), 67–74. [29] [H2]U. Hamenstädt, Anosov flows which are uniformly expanding at periodic points, à paraître. · Zbl 0805.58049 [30] [HoR]D. Hong, S. Rajeev, Universal Teichmüller space and Diff(S 1)/S 1, Comm. Math. Phys. 135 (1991), 401–411. · Zbl 0727.30037 [31] [Hop]H. Hopf, Abbildungsklassenn-dimensionaler mannigfaltigkeiten, Math. Annalen 96 (1926), 209–224. · JFM 52.0569.06 [32] [K]A. Katok, Four applications of conformal equivalence to geometry and dynamics, Ergodic Theory Dyn. Sys. 8 (1988), 139–152. · Zbl 0668.58042 [33] [KKnWe]A. Katok, G. Knieper, H. Weiss, Regularity of topological entropy, à paraître. [34] [Ki]A. Kirillov, Kähler structures onK-orbits of the group of diffeomorphisms of a circle, Funkts. Anal. Prilozhen 21:2 (1987), 42–45. · Zbl 0653.26012 [35] [KiY]A. Kirillov, D. Yuriev, Kähler of the infinite dimensional homogenous manifold Diff+(S 1)/Rot(S 1), Funct. Anal. Appl. 20 (1986), 322–324, ibid Funct. Anal. Appl. 21 (1987), 284–294. · Zbl 0644.53060 [36] [Le]C. LeBrun, Einstein metrics and Mostow rigidity, Preprint SUNY at Stony Brook (november 1994) [37] [Led1]F. Ledrappier, Harmonic measures and Bowen-Margulis measures, Israel J. Math. 71 (1990), 275–287. · Zbl 0728.53029 [38] [Led2]F. Ledrappier, Applications of dynamics to compact manifolds of negative curvature, Conférence Congrès International de Zürich (1994). [39] [Li]A. Lichnerowicz, Sur les espaces riemanniens complètement harmoniques, Bull. Soc. Math. France 72 (1944), 146–169. · Zbl 0060.38506 [40] [M]A. Manning, Topological entropy for geodesic flows, Ann. Math. 110 (1979), 567–573. · Zbl 0426.58016 [41] [Ma]G.A. Margulis, Applications of ergodic theory to the investigation of manifolds of negative curvature, Funct. Anal. Appl. 3 (1969), 335–336. · Zbl 0207.20305 [42] [Mo]F. Morgan, Calibrations and new singularities in area minimizing surfaces: a survey, Proc. Conf. Problèmes Variationnels, Paris (1988). [43] [Mos]G.D. Mostow, Quasiconformal mappings inn-space and the rigidity of hyperbolic space forms, Publications de l’IHES 34 (1968), 53–104. · Zbl 0189.09402 [44] [N]S. Nag, On the tangent space to the universal Teichmüller space, Ann. Acad. Scient. Fennicae, Seires A.I, Math. 18 (1993), 377–393. · Zbl 0794.32020 [45] [O]J.P. Otal, Le spectre marqué des longueurs des surfaces à courbure négative, Ann. Math. 131 (1990), 151–162. · Zbl 0699.58018 [46] [P]O. Pekonen, Universal Teichmüller space in geometry and physics, preprint (1993). · Zbl 0817.30022 [47] [Sc]L. Schwartz, Radon Measures on Arbitrary Topological Spaces and Cylindrical Measures, Tata Institute of Fundamental Research, Oxford University Press (1973), [48] [Se]B. Sevennec, Conversations scientifiques. [49] [Si]Y.T. Siu, The complex analyticity of harmonic maps and the strong rigidity of compact Kähler manifolds, Ann. of Math. 112 (1980), 73–111. · Zbl 0517.53058 [50] [Sp]E.H. Spanier, Algebraic Topology, McGraw-Hill (1966). [51] [Sz]Z. Szabo, The Lichnerowicz conjecture on harmonic manifolds, J. Differential Geom. 31 (1990), 1–28. [52] [T]W. Thurston, The geometry and topology of three-manifolds, Princeton University (1979). · Zbl 0409.58001 [53] [Wi]E. Witten, Coadjoint orbits and the Virasoro group, Comm. Math. Phys. 114 (1988), 1–53. · Zbl 0632.22015 [54] [Z]R.J. Zimmer, Ergodic Theory and Semisimple Groups, Monographs in Mathematics 81, Birkhäuser (1984). · Zbl 0571.58015
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