A compactness result for $\mathcal{H}$‑holomorphic curves in symplectizations

An index inequality for embedded pseudoholomorphic curves in symplectizations

Let be a surface with a symplectic form, let φ be a symplectomorphism of , and let Y be the mapping torus of φ. We show that the dimensions of moduli spaces of embedded pseudoholomorphic curves in R×Y , with cylindrical ends asymptotic to periodic orbits of φ or multiple covers thereof, are bounded from above by an additive relative index. We deduce some compactness results for these moduli spa...

Properties of Pseudoholomorphic Curves in Symplectizations Iii: Fredholm Theory

We shall study smooth maps ~ u : _ S ! RM of nite energy deened on the punctured Riemann surface _ S = S n? and satisfying a Cauchy-Riemann type equation T ~ u j = J(~ u) T ~ u for special almost complex structures J, related to contact forms on the compact three manifold M. Neither the domain nor the target space are compact. This diiculty leads to an asymptotic analysis near the punctures. A ...

A compactness result for Landau state in thin-film micromagnetics

A Density Result for Real Hyperelliptic Curves

Let {∞+,∞−} be the two points above ∞ on the real hyperelliptic curve H : y = (x − 1) ∏2g i=1(x− ai). We show that the divisor ([∞ ]− [∞−]) is torsion in Jac J for a dense set of (a1, a2, . . . , a2g) ∈ (−1, 1). In fact, we prove by degeneration to a nodal P that an associated period map has derivative generically of full rank.

Holomorphic curves in Exploded Torus Fibrations: Compactness

The category of exploded torus fibrations is an extension of the category of smooth manifolds in which some adiabatic limits look smooth. (For example, the type of limits considered in tropical geometry appear smooth.) In this paper, we prove a compactness theorem for (pseudo)holomorphic curves in exploded torus fibrations. In the case of smooth manifolds, this is just a version of Gromov’s com...