Measured J-holomorphic Foliations

Let Σ denote an oriented surface. The Narisham-Sheshadri theorem, suitably translated, says that on S²×Σ for any integrable complex structure j which makes the sections 0×Σ and ∞×Σ holomorphic there is a 1 complex dimensional foliation such that the leaves are j-invariant and the foliation has an invariant metric.

We show that this does not hold in the almost complex category. We produce a j on S²×Σ where there can be no such foliation, by proving a uniqueness theorem for foliations with smooth invariant measures. Then we prove an existence theorem for a less regular, but analagous object via an extension of Sullivan's Hahn Banach alternative.

We show that when an almost complex structure admits sufficient symmetry the objects we produce are foliations with smooth invariant measures and thus unique. However in general this uniqueness depends subtly on the regularity of the resulting “foliation”. Finally we show that constructions relating J-holmorphic curves and almost complex structures on rational surfaces admit generalizations to products of surfaces in this context.