  We consider a general class of parametrized displacement boundary value
problems in incompressible nonlinear elasticity. We prove the existence of an
unbounded solution branch of classical injective solutions emanating from the
unforced stress-free reference configuration, which constitutes a sharp global
implicit function theorem. The ramifications of this are: There exists at least
one solution for any given applied loading, and or there exists solutions of
arbitrarily large norm for some finite loading. We refer to this as solutions
in the large. The nonlinear constraint equation enforcing incompressibility
obviates the direct use of either the Leray-Schauder degree or an oriented
degree previously developed for compressible problems. Instead we employ the
nonlinear Fredholm degree for proper maps developed by Fitzpatrick, et. al. We
establish the requisite admissibility properties in the presence of the
constraint equation before proving the main results.
