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

Let $(M,g)$ be an asymptotically conical Riemannian manifold having dimension n≥ 2, opening angle α ∈ (0,π/2) \ 1/2k+1 ∈ 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 ∂_∞ M = Sⁿ⁻¹, we prove the existence of a geodesic line with asymptotic prescribed boundaries and the Morse index ≤ n-1.