We prove a certain non-linear version of the Levi extension theorem for
meromorphic functions. This means that the meromorphic function in question is
supposed to be extendable along a sequence of complex curves, which are
arbitrary, not necessarily straight lines. Moreover, these curves are not
supposed to belong to any finite dimensional analytic family. The conclusion of
our theorem is that nevertheless the function in question meromorphically
extends along an (infinite dimensional) analytic family of complex curves and
its domain of existence is a pinched domain filled in by this analytic family.Comment: 19 pages, This is the final version with significant corrections and
improvements. To appear in Arkiv f\"or matemati
