Finite-energy pseudoholomorphic planes with multiple asymptotic limits

Abstract

It’s known from (Ann Inst H Poincaré Anal Non Linéaire 13(3):337–379 [11], Properties of Pseudoholomorphic Curves in Symplectisation. IV. Asymptotics with Degeneracies. In: Contact and symplectic geometry (Cambridge, 1994), vol 8 Publ. Newton Inst., pp 78–117. Cambridge Univ. Press, Cambridge [10], A Morse-Bott Approach to Contact Homology. PhD thesis, Stanford University [2]) 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 nonremovable 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.