One Dimensional Asymptotic Plateau Problem in $n$-Dimensional Asymptotically Conical Manifolds

Let $(M,g)$ be an asymptotically conical Riemannian manifold having dimension $n\ge 2$, opening angle $\alpha \in (0,\pi/2) \setminus \{\arcsin \frac{1}{2k+1}\}_{k \in \mathbb{N}}$ and positive asymptotic rate. Under the assumption that the exponential map is proper at each point, we give a solution to the one dimensional asymptotic Plateau problem on $M$. Precisely, for any pair of antipodal points in the ideal boundary $\partial_\infty M = \mathbb S^{n-1}$, we prove the existence of a geodesic line with asymptotic prescribed boundaries and the Morse index $\le n-1$.