Failure of parametric H-principle for maps with prescribed Jacobian

Let M and N be closed n-dimensional manifolds, and equip N with a volume form σ. Let μ be an exact n-form on M. Arnold then asked the question: When can one find a map f:M→N such that f^∗σ=μ. In 1973 Eliashberg and Gromov showed that this problem is, in a deep sense, trivial: It satisfies an h-principle, and whenever one can find a bundle map f_{b d l}:T M→T N which is degree 0 on the base and such that f_{b d l}^∗(σ)=μ one can homotop this map to a solution f. That is if the naive topological conditions are satisfied on can find a solution. There is no further interesting geometry in the problem.

We show the corresponding parametric h-principle fails- if one considers families of maps inducing μ from σ, one can find interesting topology in the space G_μ of solutions which is not predicted by an h-principle. Moreover the homotopy type of such maps is “quantized”: for certain families of forms homotopy type remains constant, jumping only at discrete values.

Symplectomorphism Groups and Isotropic Skeletons

(Accepted by Geometry and Topology) The symplectomorphism group of a 2-dimensional surface S is homotopy equivalent to the orbit of a filling system of curves. We give a generalization of this statement to dimension 4. The filling system of curves is replaced by the Biran decomposition of the symplectic 4 manifold M into a disjoint union of an isotropic 2-complex L and a disc bundle over a symplectic surface \Sigma, Poincare dual to a multiple of the form. We show that one can recover the homotopy type of the symplectomorphism group of M from the orbit of the pair (L,\Sigma). This allows us to compute the homotopy type of certain spaces of Lagrangian submanifolds, for example the space of Lagrangian RP^2 in CP^2 which are isotopic to the standard one.math.SG/0404496:

Maps with symplectic graphs

(Submitted to Commentari Math Helv.) We consider the homotopy type of maps between symplectic surface whose graphs form symplectic submanifolds of the product. We give a purely topological model for this space in terms of maps with constrained numbers of pre-images. We use this to show that the dependence of the homotopy type on the area forms of each surface is quantized- it changes only when the parameters pass certain discrete levels. When the domain is a sphere or torus, and its total area is smaller than the range, we compute the full homotopy type of the low degree components. We also give an example, showing that the homotopy type of the space of sympllectic sections of a symplectic fibration F must sometimes change as we deform F. Much of this work generalizes to n-dimensional manifolds equipped with volume forms.

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.