Finite-energy pseudoholomorphic planes with multiple asymptotic limits

TL;DR

This paper demonstrates that in degenerate contact forms, finite-energy pseudoholomorphic curves can have multiple asymptotic limits, contrasting with the unique limit behavior in nondegenerate cases.

Contribution

It provides explicit examples showing that degeneracy in contact forms allows pseudoholomorphic curves to have multiple asymptotic limits, challenging previous assumptions.

Findings

Existence of pseudoholomorphic cylinders with embedded tori as limits

Contrasts with exponential convergence in nondegenerate cases

Degeneracy permits multiple asymptotic limits for finite-energy curves

Abstract

It's known from from work of Hofer, Wysocki, and Zehnder [1996] and Bourgeois [2002] that in a contact manifold equipped with either a nondegenerate or Morse-Bott contact form, a finite-energy pseudoholomorphic curve will be asymptotic at each of its non removable punctures to a single periodic orbit of the Reeb vector field and that the convergence is exponential. We provide examples here to show that this need not be the case if the contact form is degenerate. More specifically, we show that on any contact manifold $(M, \xi)$ with cooriented contact structure one can choose a contact form $\lambda$ with $\ker\lambda=\xi$ and a compatible complex structure $J$ on $\xi$ so that for the associated $\mathbb{R}$-invariant almost complex structure $\tilde J$ on $\mathbb{R}\times M$ there exist families of embedded finite-energy $\tilde J$-holomorphic cylinders and planes having embedded tori as limit sets.