Every oriented –manifold
admits a stable folded symplectic structure, which in turn determines a homotopy
class of compatible almost complex structures that are discontinuous across the
folding hypersurface (“fold”) in a controlled fashion. We define folded holomorphicmaps, ie pseudoholomorphic maps that are discontinuous across the fold. The
boundary values on the fold are mediated by tunneling maps which are punctured
–holomorphic
maps into the folding hypersurface with prescribed asymptotics on closed
characteristics.
Our main result is that the linearized operator of this boundary value problem is
Fredholm, under the simplifying assumption that we have circle-invariant
folds.
As examples we characterize the moduli space of maps into the folded elliptic fibration
and we construct
examples of degree
rational maps into .
Moreover we explicitly give the moduli space of degree 1 rational maps into
and
show that it possesses a natural compactification.
This aims to generalize the tools of holomorphic maps to all oriented
–manifolds
by utilizing folded symplectic structures rather than other types of pre-symplectic
structures as initiated by Taubes.
Keywords
pseudoholomorphic curves, boundary value problems on
manifolds, folded symplectic structures
