  Despite their potential importance for understanding astrophysical jets,
physically realistic exact solutions for magnetospheres around Kerr black holes
have not been found, even in the force-free approximation. Instead approximate
analytical solutions such as the Blandford-Znajek (split-)monopole, as well as
numerical solutions, have been constructed. In this paper we consider a new
approach to the analysis and construction of such magnetospheres. We consider
force-free electrodynamics close to the rotation axis of a magnetosphere
surrounding a Kerr black hole assuming axisymmetry. This is the region where
the force-free approximation should work the best, and where the jets are
located. We perform a systematic study of the asymptotic region with
(split-)monopole, paraboloidal and vertical asymptotic behaviors. Imposing
asymptotics similar to a (split-)monopole, we find under certain assumptions
that demanding regularity at the rotation axis and the event horizon restricts
solutions of the stream equation so much that it is not possible for a solution
to be continuously connected to the static (split-)monopole around the
Schwarzschild black hole in the limit where the rotation goes to zero. On the
one hand, this result provides independent evidence to the issues discovered
with the asymptotics of the Blandford-Znajek (split-)monopole in Ref. [1]. On
the other hand, we also point out possible caveats in our arguments that one
could conceivably exploit to amend the perturbative construction of the
Blandford-Znajek (split-)monopole.
