Asymptotically hyperbolic manifolds with boundary conjugate points but no interior conjugate points (Q2050525)

Let \(\overline{M}\) be a smooth compact manifold with boundary whose interior is \(M\). Let \(r:\partial M\rightarrow \mathbb{R}\) be a smooth function such that \(\mid dr\mid_{\bar{g}}=1\) on \(\partial M\). A Riemannian metric \(g\) on \(M\) is called asymptotically hyperbolic (AH) if \(\bar{g}=r^{2}g\) extends smoothly to a Riemannian metric on \(\overline{M}\). It was proved in [\textit{B. Czech} et al., J. High Energy Phys. 2015, No. 10, Paper No. 175, 41 p. (2015; Zbl 1388.83217)] that AH metrics are complete and its sectional curvature approaches \(-1\) as \(r\to 0\). Such manifolds are called asymptotically hyperbolic manifolds (AHM). The Poincaré model of the hyperbolic space is the standard model where the underlying space is the n-ball \(B=\{x\in\mathbb{R}^{n+1};\ \mid x\mid<1\}\) and \(g=4\frac{\sum dx^{2}_{j}}{(1-\mid x\mid^{2})^{2}}\). The main issue concerns the existence of conjugated points on a geodesic \(\gamma\), which is equivalent to the existence of a Jacobi field vanishing at least twice along \(\gamma\). The article considers the following cases: the points \(p^{+},p^{-}\in\partial \overline{M}\) are boundary conjugate if there exists a geodesic \(\gamma\) such that (i) \(\lim_{t\to\pm\infty}\gamma(t)=p^{\pm}\) and (ii) there exists a Jacobi field \(Y\) along \(\gamma\) satisfying \(\lim_{t\to\pm\infty}\mid Y(t)\mid=0\). When \(p^{+},p^{-}\in M\) and conditions (i) and (ii) are satisfied they are interior conjugated points. As we should expect, on an (AHM) manifold with non-positive sectional curvature there are no interior or boundary conjugate points. In [J. Inverse Ill-Posed Probl. 5, No. 6, 487--490 (1997; Zbl 0908.35136)] \textit{Yu. E. Anikonov} and \textit{V. G. Romanov} proved that if an (AHM) manifold \(M\) has no interior conjugated points, then there is no Jacobi field \(Y\) vanishing in \(M\) and at \(\partial \overline{M}\), i.e., there are no interior-boundary conjugate points. An (AHM) is called non-trapping when given any compact set \(K\subset M\) and a unity speed geodesic \(\gamma(t)\) there exists \(T_{K,\gamma}\subset M\) so that \(\gamma(t)\notin T_{K,\gamma}\) for all \(t\) such that \(\mid t\mid\ge T_{K,\gamma}\). In [1], it was proved that any geodesic of a non-trapping (AHM) approaches the boundary as \(t\to\pm\infty\). \textbf{Main Theorem}. For any integer \(n\ge 1\) there exist smooth non-trapping asymptotically hyperbolic manifolds of dimension \((n+1)\) with boundary conjugate points but no interior conjugate points. The author stress that their interest on the subject arose in connection with the formulation of the definition of a simple asymptotically hyperbolic manifold (SAHM) given in [\textit{C. A. Berenstein} and \textit{E. C. Tarabusi}, Duke Math. J. 62, No. 3, 613--631 (1991; Zbl 0742.44002)]. A simple manifold (SM) is a compact Riemannian manifold \(\overline{M}\) with a strictly convex boundary \(\partial \overline{M}\ne \varnothing\), non-trapping, and such that no pair of points in \(\overline{M}\) is composite of conjugate points in \(\overline{M}\). These manifolds are basic structures for the study of geometric inverse problems, where the study of geodesic X-ray transform plays a important role. Following the authors, they say that ``the paper [loc. cit.] was concerned with extending the study of X-ray transform to (AHM) setting, which requires the concept of a simple (AHM). In the (AHM) case, the boundary convexity comes for free since for any function \(r:\partial{\overline{M}}\rightarrow\mathbb{R}\) and \(\epsilon>0\) small enough the sets \(r\ge \epsilon\) are strictly convex''. The definition in [loc. cit.] of a (SAHM) is just the conditions to be a (AHM) plus the conditions to be a (SM). However, the conditions defining a (SAHM) imply there are no interior conjugate points either. So, it is natural to ask if it is equivalent to assume the manifold is just (AH), non-trapping and without interior conjugate points, i.e., ignoring the boundary strictly convexity. Their main theorem claims there is no such equivalence. Following their lines, the construction of a manifold to prove the main theorem relies on a technique developed in [\textit{X. Chen} and \textit{A. Hassell}, Commun. Partial Differ. Equations 41, No. 3, 515--578 (2016; Zbl 1351.58018)] to construct metrics with the desired properties and elucidating the relationship among the existence of conjugate points, Anosov geodesic flows, existence of focal points, and existence of open sets of strictly positive curvature. They start constructing a complete, non-trapping and \(O(n+1)\)-invariant Riemannian metric on \(\mathbb{R}^{n+1}\) of class \(C^{1,1}\) which compactifies to an (AH) metric, such that there are no nontrivial Jacobi fields vanishing twice in the interior but along radial geodesics there are Jacobi fields vanishing as both \(t\to\pm\infty\).