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Contact homology and higher dimensional closing lemmas

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  • We develop methods for studying the smooth closing lemma for Reeb flows in any dimension using contact homology. As an application, we prove a conjecture of Irie, stating that the strong closing lemma holds for Reeb flows on ellipsoids. Our methods also apply to other Reeb flows, and we illustrate this for a class of examples introduced by Albers–Geiges–Zehmisch.

    Mathematics Subject Classification: Primary: 53E50; Secondary: 53D42.

    Citation:

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  • Figure 1.  Morally speaking, the strong closing property states that any positive perturbation of a Reeb flow supported in $ U $ must produce a closed orbit $ \gamma $ through $ U $

    Figure 2.  A cartoon of a U-map curve in contact homology. In general, the tangency of the curve $ C $ at the sub-manifold $ \Sigma $ can be arbitrarily complex

    Figure 3.  Two different visualizations of the Reeb flow of an ellipsoid $ E \subset \mathbb{C}^n $. On the left, a flow on the boundary of the higher-dimensional domain $ E $. On the right, as the dynamics of $ n $ independent harmonic oscillators (e.g., springs)

    Figure 4.  A simplified picture of a possible holomorphic building. Note that all of the holomorphic buildings and curves of interest in this paper will be genus $ 0 $

    Figure 5.  A cartoon of (a) an abstract constraint $ P $, visualized as a weighting of the orbits on the sphere and (b) the curves counted in constrained cobordism maps, asymptotic to the orbits with non-zero weights under $ P $

    Figure 6.  An illustration of a contact flag, as defined in Setup 4.1. The lavender sphere $ Y_7 $, the pink disk $ Y_5 $ and the purple circle $ Y_3 $ represent consecutive nested sub-manifolds in the flag. The orbits $ \gamma_+ $ and $ \gamma_- $ lie in $ Y_3 $, and are connected by a Morse flow line in green

    Figure 8.  An illustration of the deformation $ u^\tau $ of $ u $, whose ends are disjoint from the hypersurface, and the cylinder $ \sigma $ that has the same ends and does not intersect $ \hat Y_{2n-3} $

    Figure 7.  An illustration of a holomorphic building $ \overline{u} $, and the sequence of orbits $ \{\gamma^j\} $

    Figure 9.  The proof of Proposition 4.9 uses positivity of intersection to rule out the illustrated scenario, in which the piece of the building that intersects the hypersurface is not contained in it

    Figure 10.  A possible building in the moduli space $ \overline{\mathcal{M}}_{X \setminus E \sqcup E} $. We will show in Lemma 4.30 that (generically) these buildings must be much simpler than this

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