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Finite-energy pseudoholomorphic planes with multiple asymptotic limits

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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.

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Notes

  1. In fact, a claimed proof that no such examples exist has appeared in a recent (now-withdrawn) arXiv preprint.

  2. Hofer only considers planes in [7] and proves the slightly weaker statement that there exists a sequence \(s_{k}\rightarrow \infty \) so that corresponding loops \(u\circ \psi (s_{k}, \cdot )\) converge to a periodic orbit, but the generalization of the proof to the result we state here is straightforward. In the survey [13, Theorem 3.2], the appropriate result is proven for a general pseudoholomorphic half-cylinder, albeit under a different notion of energy. The fact that this different notion of energy implies finite Hofer energy as defined by (8) is addressed in Theorem 5.1 of the same paper.

  3. We remark that finiteness of the energy here does not immediately imply that the cylinders approach periodic orbits because W, being equipped with an exact symplectic form, is necessarily noncompact. Consider, for example, the symplectic manifold \((\mathbb {R}^{2}, dx\wedge dy=d(x\,dy))\) and the function \(f(x, y)=\arctan x\). The pseudoholomorphic cylinders in the appropriate prequantization space covering gradient flow lines in the base have finite energy as a result of (18) since the function f is bounded, but the cylinders do not approach periodic orbits since the function f has no critical points.

  4. As will become clear from our proof, the 5 / 4 in our example can be replaced with any constant strictly bigger than 1 and strictly less than \(\sqrt{2}\).

  5. The left-most part of this inequality can be seen from the following argument. To show that \(-\frac{9}{4}e^{1/s}-s\) is positive for all \(s<0\), it suffices to show that \(g(t)=\frac{9}{4}te^{t}+1\) is positive for all \(t<0\). A straightforward argument using single-variable calculus then shows that \(g(-1)=-\frac{9}{4}e^{-1}+1>0\) is the absolute minimum of the function g on \(\mathbb {R}\).

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Acknowledgments

I would like to thank Luis Diogo and Urs Frauenfelder for helpful discussions. I also gratefully acknowledge financial support from DFG grant BR 5251/1-1 and the Fakultät für Mathematik at the Ruhr-Universität Bochum.

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  1. Fakultät für Mathematik, Ruhr-Universität Bochum, 44780, Bochum, Germany

    Richard Siefring

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Correspondence to Richard Siefring.

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Communicated by Jean-Yves Welschinger.

Dedicated to Helmut Hofer on the occasion of his 60th birthday, and in warm remembrance of Kris Wysocki.

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Siefring, R. Finite-energy pseudoholomorphic planes with multiple asymptotic limits. Math. Ann. 368, 367–390 (2017). https://doi.org/10.1007/s00208-016-1478-y

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