Asymptotics of curves prescribed on Hadamard manifolds (Q1320022)

In the Euclidean plane a parabola parametrized by arc length has curvature behaving like (constant) / \(s^{3/2}\) as \(s\to\infty\) while a hyperbola has curvature behaving like (constant) / \(s^ 3\) as \(s\to\infty\). In the case of the parabola there is an asymptotic direction as \(s\to\infty\), which is the case for any curve whose curvature is \(O(1/s^{1+\varepsilon})\) for some \(\varepsilon>0\) as \(s\to\infty\). The hyperbola has an asymptotic line as \(s\to\infty\), which is the case for any curve with curvature \(O(1/s^{2+\varepsilon})\) as \(s\to\infty\). In this article, the authors generalize these results to Hadamard manifolds \(\widetilde M\): complete, simply connected Riemannian manifolds with sectional curvature \(K\leq 0\). Let \(\widetilde M(\infty)\) denote the boundary of \(\widetilde M\) consisting of equivalence classes of asymptotic unit speed geodesics of \(\widetilde M\). The space \(\overline M= \widetilde M\cup \widetilde M(\infty)\) admits two natural topologies, the cone and horosphere topologies. Let \(\alpha(t)\) denote a unit speed curve in \(\widetilde M\) and let \(k_ g(t)\) denote its geodesic curvature. The authors consider the following two conditions: (H1) There exists \(\varepsilon> 0\) such that \(k_ g(t)= O(1/t^{1+\varepsilon})\) as \(t\to\infty\). (H2) The same condition with \(1+\varepsilon\) replaced by \(2+\varepsilon\). Theorem 1. If \(\alpha(t)\) is a curve in \(\widetilde M\) satisfying (H1), then \(\alpha(t)\) is unbounded in \(\widetilde M\) and converges to a point \(x\) in \(\widetilde M(\infty)\) as \(t\to\infty\) with respect to both the cone and horosphere topologies. If \(\gamma\) is a geodesic belonging to \(x\), then \(d(\alpha(t),\gamma)= o(t)\) as \(t\to\infty\). If (H2) holds for \(\alpha(t)\), then \(d(\alpha(t),\gamma)\leq c\) for some \(c>0\) and all \(t\geq 0\). The geodesic \(\gamma\) above that belongs to \(x\) is called an asymptote of \(\alpha(t)\) in the case that (H2) holds. If \(d(\alpha(t),\gamma)\to 0\) as \(t\to\infty\), then \(\gamma\) is called a strict asymptote. Given a point \(x\) in \(\widetilde M(\infty)\), the space \(G(x)\) of geodesics belonging to \(x\) becomes a metric space with metric \(d(\gamma,\sigma)= \inf\{d(\gamma(s), \sigma(t): s,t\in \mathbb{R}\}\) for geodesics \(\gamma\), \(\sigma\) belonging to \(x\). We identify any two geodesics \(\gamma\), \(\sigma\) of \(x\) if \(d(\gamma,\sigma)=0\). Theorem 2. Let \(\alpha(t)\) be a curve in \(\widetilde M\) satisfying (H2), and let \(\alpha(t)\) converge to \(x\) in \(\widetilde M(\infty)\) as \(t\to\infty\). If the metric space \(G(x)\) is complete, then \(\alpha(t)\) admits a strict asymptote \(\gamma\). We say \(\widetilde M\) has curvature order at most 2 at a point \(x\) in \(\widetilde M(\infty)\) if for any geodesic \(\gamma\) belonging to \(x\) the integral \(\int^ \infty_ 0 t k(\gamma'(t)) dt=+ \infty\), where \(k(\gamma'(t))\) denotes the minimum absolute value of the sectional curvatures of those 2-planes containing \(\gamma'(t)\). If \(\widetilde M\) has curvature order at most 2 at a point \(x\) in \(\widetilde M(\infty)\), then \(G(x)\) is a single point. The authors give an example to show that Theorem 2 is false if the space \(G(x)\) is not complete. The article also studies the spaces \(G(x)\), \(x\in \widetilde M(\infty)\), in the case that \(\widetilde M\) has dimension 2 and particularly in the case that \(\widetilde M\) arises from a complex algebraic curve in \(\mathbb{C}^ 2\).