Free boundary minimal hypersurfaces of Morse index one in Eucildean half-space

We prove that a complete, connected, two-sided, embedded free boundary minimal hypersurface of finite total curvature and Morse index one in a half-space of RN, with nonempty boundary is a higher-dimensional half-catenoid. We also prove that a complete, connected, two-sided, embedded free boundary minimal hypersurface of finite total curvature and Morse index one in a quarter-space is the higher-dimensional quarter-catenoid.