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The structure of steady axisymmetric force-free magnetosphere of a Kerr blackhole (BH) is governed by a second-order partial differential equation of$A_\phi$ depending on two "free" functions $\Omega(A_\phi)$ and $I(A_\phi)$,where $A_\phi$ is the $\phi$ component of the vector potential of theelectromagnetic field, $\Omega$ is the angular velocity of the magnetic fieldlines and $I$ is the poloidal electric current. In this paper, we investigatethe solution uniqueness. Taking asymptotically uniform field as an example,analytic studies imply that there are infinitely many solutions approachinguniform field at infinity, while only a unique one is found in generalrelativistic magnetohydrodynamic simulations. To settle down the disagreement,we reinvestigate the structure of the governing equation and numerically solveit with given constraint condition and boundary condition. We find that theconstraint condition (field lines smoothly crossing the light surface (LS)) andboundary conditions at horizon and at infinity are connected via radiationconditions at horizon and at infinity, rather than being independent. Withappropriate constraint condition and boundary condition, we numerically solvethe governing equation and find a unique solution. Contrary to naiveexpectation, our numerical solution yields a discontinuity in the angularvelocity of the field lines and a current sheet along the last field linecrossing the event horizon. We also briefly discuss the applicability of theperturbation approach to solving the governing equation.

Analytic properties of force-free jets in the Kerr spacetime -- III: uniform field solution