Mostow rigidity and the Bishop-Steger dichotomy for surfaces of variable negative curvature
The author finds necessary and sufficient conditions for two compact surfaces of negative Gaussian curvature to have the same marked length spectrum. The first result stated below generalizes a result of Bishop and Steger for compact surfaces of constant negative curvature. If \((M,g)\) denotes a compact Riemannian manifold of sectional curvature \(K<0\), then each free homotopy class \(\beta_ \delta\) of closed curves in \(M\) admits a unique closed geodesic of length \(\lambda_ g(\beta_ \delta)\). The collection of pairs \(\{(\beta_ \delta,\lambda_ g(\beta_ \delta))\}\), as \(\delta\) ranges over the set \(\Delta\) of free homotopy classes of \(M\), is called the marked length spectrum of \((M,g)\). If two compact negatively curved manifolds \((M_ 1,g_ 1)\) and \((M_ 2,g_ 2)\) have isomorphic fundamental groups, then there is a natural bijection between the free homotopy classes of \(M_ 1\) and \(M_ 2\). If this bijection preserves the lengths of the closed geodesics in corresponding free homotopy classes, then \((M_ 1,g_ 1)\) and \((M_ 2,g_ 2)\) are said to have the same marked length spectrum. \textit{C. Croke} [Comment. Math. Helv. 65, 150-169 (1990; Zbl 0704.53035)] has shown that two compact surfaces of nonpositive Gaussian curvature with the same marked length spectrum are isometric. \textit{U. Hamenstädt} in [Ergodic Theory Dyn. Syst. 12, No. 1, 67-74 (1992; Zbl 0766.58045)] has shown that two compact manifolds with \(K<0\) and arbitrary dimension have the same marked length spectrum if and only if there is a homeomorphism between the unit tangent bundles that conjugates the geodesic flows in time preserving fashion. In stating the results of this article we let \(M\) denote a compact surface with Riemannian metrics \(g_ 1\), \(g_ 2\) of negative curvature. For \(i=1,2\), let \((\widetilde M,\widetilde g_ i)\) denote the universal cover of \((M,g_ i)\), and let \(\widetilde d_ i\) denote the Riemannian distance function in \((\widetilde M,\widetilde g_ i)\). Theorem 1. The following statements are equivalent: 1) \((M,g_ 1)\) and \((M,g_ 2)\) have the same marked length spectrum. 2) Let \(\Delta\) denote the set of free homotopy classes of \(M\). For each \(0<s<1\) one has \(\sum_{\delta\in\Delta} \exp\{-\lambda_ \delta(s)\}=\infty\), where \(\lambda_ \delta(s)= s\lambda_{g_ 1}(\beta^ 1_ \delta)+ (1-s) \lambda_{g_ 2}(\beta^ 2_ \delta)\). 3) Let \(\Gamma\) denote the fundamental group of \(M\) acting as a discrete group of isometries of the universal cover \(\widetilde M\). Then for each \(x\in\widetilde M\) and \(0<s<1\) we have \(\sum_{\gamma\in\Gamma}\exp\{-\delta_ \gamma(s,x)\}=\infty\), where \(\delta_ \gamma(s,x)= s \widetilde d_ 1(x,\gamma x)+ (1-s)\widetilde d_ 2(x,\gamma x)\). Moreover, if 1), 2) and 3) do not hold, then for every \(0<s<1\) there exists \(0< p<1\) such that for all \(x\in\widetilde M\) we have \(\sum_{\delta\in\Delta} \exp\{- p \lambda_ \delta(s)\}<\infty\) and \(\sum_{\gamma\in\Gamma}\exp\{- p \delta_ \gamma(s,x)\}<\infty\). If \(M\) is a compact surface with \(K<0\), then \(\widetilde M\) admits natural Liouville and Patterson measure classes, \(\nu^ L\) and \(\nu\), which consist of families of measures \(\{\nu^ L_ x: x\in\widetilde M\}\) and \(\{\nu_ x: x\in\widetilde M\}\) such that any two members of a family are absolutely continuous with respect to each other. The author also proves the following Theorem 2. The following statements are equivalent: 1) \((M,g_ 1)\) and \((M,g_ 2)\) hae the same marked length spectrum. 2) The Liouville measure classes of \((\widetilde M,g_ 1)\) and \((\widetilde M,g_ 2)\) are the same. 3) The Patterson measure classes of \((\widetilde M,g_ 1)\) and \((\widetilde M,g_ 2)\) are the same. Moreover, if 1), 2) and 3) do not hold, then the Liouville measure classes of \((\widetilde M,g_ 1)\) and \((\widetilde M,g_ 2)\) are mutually singular and the Patterson measure classes of \((\widetilde M,g_ 1)\) and \((\widetilde M,g_ 2)\) are mutually singular.
scientific article; zbMATH DE number 179214
scientific article