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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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