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Symplectic Non-hyperbolicity

Spencer Cattalani

TL;DR

This paper develops a symplectic framework for complex lines using Ahlfors currents, extending Bangert's existence results and connecting holomorphic disk dynamics to non-hyperbolicity in the symplectic category. It builds a toolbox—filling functions, relative fillings, large-scale monotonicity, and a continuity method—to guarantee the existence of Ahlfors currents and to analyze their qualitative behavior. A bubbling theorem via averaging shows that families of $J$-holomorphic curves yield Ahlfors currents as weak limits, with corollaries including GW-type invariants under suitable compactness. It also proves that the space of Ahlfors currents is convex and weakly compact, enabling a geometric understanding of their extreme points and suggesting new symplectic invariants derived from their convex geometry. Altogether the results quantify transcendental noncompactness phenomena in symplectic geometry and provide a robust framework for studying symplectic non-hyperbolicity through Ahlfors currents.

Abstract

Complex (affine) lines are a major object of study in complex geometry, but their symplectic aspects are not well understood. We perform a systematic study based on their associated Ahlfors currents. In particular, we generalize (by a different method) a result of Bangert on the existence of complex lines. We show that Ahlfors currents control the asymptotic behavior of families of pseudoholomorphic curves, refining a result of Demailly. Lastly, we show that the space of Ahlfors currents is convex.

Symplectic Non-hyperbolicity

TL;DR

This paper develops a symplectic framework for complex lines using Ahlfors currents, extending Bangert's existence results and connecting holomorphic disk dynamics to non-hyperbolicity in the symplectic category. It builds a toolbox—filling functions, relative fillings, large-scale monotonicity, and a continuity method—to guarantee the existence of Ahlfors currents and to analyze their qualitative behavior. A bubbling theorem via averaging shows that families of -holomorphic curves yield Ahlfors currents as weak limits, with corollaries including GW-type invariants under suitable compactness. It also proves that the space of Ahlfors currents is convex and weakly compact, enabling a geometric understanding of their extreme points and suggesting new symplectic invariants derived from their convex geometry. Altogether the results quantify transcendental noncompactness phenomena in symplectic geometry and provide a robust framework for studying symplectic non-hyperbolicity through Ahlfors currents.

Abstract

Complex (affine) lines are a major object of study in complex geometry, but their symplectic aspects are not well understood. We perform a systematic study based on their associated Ahlfors currents. In particular, we generalize (by a different method) a result of Bangert on the existence of complex lines. We show that Ahlfors currents control the asymptotic behavior of families of pseudoholomorphic curves, refining a result of Demailly. Lastly, we show that the space of Ahlfors currents is convex.

Paper Structure

This paper contains 7 sections, 28 theorems, 51 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Existence of Ahlfors currents
  3. Filling functions
  4. Large scale monotonicity
  5. Continuity method
  6. Bubbling Ahlfors currents
  7. Flexibility of disks

Key Result

Theorem 1.1

There exists a nonconstant $J$-holomorphic map $u: \mathbb C \longrightarrow T^{2n}$ for any almost complex structure $J$ tamed by the standard symplectic structure on $T^{2n}$.

Figures (4)

  • Figure 1: The geodesic cone over a curve centered at a point, as in Example \ref{['nonpositive_filling_function_eg']}, is depicted on the left. The geodesic cylinder over a curve based on a totally geodesic submanifold, as in Example \ref{['totally_geodesic_eg']}
  • Figure 2: The chain $C$ filling $\gamma_1 + \gamma_2$ in the proof of Lemma \ref{['superadditivity_lmm']}
  • Figure 3: The shaded region is the chain $C$ constructed in the proof of Lemma \ref{['relative_from_filling_lmm']}.
  • Figure 4: The shaded area is the disk constructed in Lemma \ref{['spectacles_lmm']}. In this figure, $m_i = 2$ and $n_i = 3$.

Theorems & Definitions (69)

  • Theorem 1.1: Ban98
  • Definition 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Corollary 1.6
  • Corollary 1.7
  • Corollary 1.8
  • Theorem 1.9
  • Corollary 1.10
  • ...and 59 more