Hausdorff approximations on Hadamard manifolds and their ideal boundaries (Q1375803)

The author considers the ideal boundary of a Hadamard manifold (a complete, simply connected Riemannian manifold \(M\) with nonpositive sectional curvature). The ideal boundary \(M(\infty)\) consists of asymptote classes of unit speed geodesics of \(M\), and for each point \(p\) of \(M\) and each point \(x\) of \(M(\infty)\) there exists a unique geodesic \(\gamma_{px}\) that belongs to \(x\) and satisfies \(\gamma_{px}(0) = p.\) There exists a natural sphere topology on \(M(\infty)\) such that the map \(x\to \gamma_{px}'(0)\) is a homeomorphism of \(M(\infty)\) onto the unit sphere in \(T_pM\) for every point \(p\) in \(M\). Gromov introduced a pseudometric \(\text{Td}\) on \(M(\infty)\), usually called the Tits metric, which may be defined as follows. Fix a point \(p\) of \(M\). For points \(z,z^*\) in \(M(\infty)\) define \(\ell(z,z^*)= \lim_{t\to\infty}\ell_t(z,z^*)\), where \(\ell_t (z,z^*)= (1/t)d(\gamma_{pz}(t),\gamma_{pz^*}(t))\), and define \(\text{Td}\) to be the inner metric on \(M(\infty)\) determined by \(\ell\), which is independent of \(p\). Even if \(\text{Td}\) is finite on \(M(\infty)\), the metric space \((M(\infty), \text{Td})\) is rarely compact; if \(M\) is a symmetric space, then \((M(\infty), \text{Td})\) is compact if and only if \(M\) is isometric to flat Euclidean space. For two metric spaces \(X,Y\) a map \(\varphi:X\to Y\) is an \((\alpha,\Delta)\) rough isometry if every point of \(Y\) lies in the closed \(\Delta\)-neighborhood of \(\varphi(X)\) and if \((1/\alpha)d(x,x^*)-\Delta\leq d(\varphi(x),\varphi(x^*)\leq \alpha d(x,x^*)+\Delta\) for all points \(x,x^*\) in \(X\). For \(t > 0\) and \(z,z^*\) in \(M(\infty)\) define \(\alpha_t(z,z^*) = \ell_t(z,z^*)/\ell(z,z^*)\) and \(\alpha_t = \inf\{\alpha_t(z,z^*): z,z^*\) are arbitrary in \(M(\infty)\) with \(z\neq z^*\}\). The numbers \(\alpha_t\) are nondecreasing in \(t\) and bounded above by 1. If \(\alpha=\lim_{t\to\infty} \alpha_t\) is positive, then the \(\text{Td}\) and sphere topologies on \(M(\infty)\) are the same, and in particular \((M(\infty),\text{Td})\) is compact. If \(\alpha= 1\), then \(M\) is said to satisfy condition (E). The author has found examples of Hadamard manifolds \(M,M^*\) that satisfy condition (E) but are not \((1,\Delta)\) rough isometric. Theorem 1. Let \(M,M^*\) be Hadamard manifolds, and let \(\varphi:M\to M^*\) by a \((1,\Delta)\) rough isometry for some \(\Delta > 0\). Then \((M(\infty),\text{Td})\) and \((M^*(\infty),\text{Td})\) are isometric. Theorem 2. Let \(M,M^*\) be Hadamard manifolds satisfying condition (E) such that \((M(\infty),\text{Td})\) and \((M^*(\infty),\text{Td})\) are isometric. Then for every \(\varepsilon > 0\) there exists a number \(T_\varepsilon > 0\) and a \((1 +\varepsilon, T_\varepsilon)\) rough isometry between \(M\) and \(M^*\).