Almost complex structures and calibrated integral cycles in contact 5-manifolds (Q2842999)

Let \((\mathcal{M}^5,\alpha)\) be a contact \(5\)-manifold, i.e. \(\alpha\wedge (d\alpha)^2\neq 0\). Denote by \(J\) an almost complex structure on the horizontal distribution \(H\) of the given contact structure. If \(R_\alpha\) is the Reeb vector field of \(\alpha\), one extends \(J\) to the entire tangent bundle by \(J(R_\alpha)=0\). Then \(J^2=-Id+R_\alpha \otimes \alpha\). If \(g_{J, d\alpha} \) is an associated Riemannian metric on the horizontal distribution \(H\) (then \(g_{J, d\alpha} (v,w)=d\alpha (v,Jw)\)), its extension \(g= g_{J, d\alpha} +\alpha\otimes \alpha\) gives a metric contact structure on \(\mathcal{M}\). For a given \(2\)-form \(\phi\) on a Riemannian manifolds \((M,g)\), the comass of \(\phi\) is defined by \(\|\phi\|_*= \sup \{<\phi_x,\xi_x>:x\in M, \xi_x \;\text{is\;a\;unit\;simple\;2-vector\;at}\;x\}\). A form \(\phi\) of comass one is called a calibration if it is closed. The author is concerned with semi-calibrated cycles; they are almost minimizers of the area functional. An extension of the results from the paper [\textit{C. Bellettini} and \textit{T. Rivière}, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 11, No. 1, 61--142 (2012; Zbl 1242.49093)] is obtained. The main result of this paper is Theorem 1.2. Let \(\mathcal{M}\) be a five-dimensional manifold endowed with a contact form \(\alpha\) and let \(J\) be an almost complex structure defined on the horizontal distribution \(H= Ker\;\alpha\), such that \(d\alpha (Jv,v)=0\) for any \(v\in H\). Let \(\mathcal{C}\) be an integer multiplicity rectifiable cycle of dimension \(2\) in \(\mathcal{M}\) such that \(\mathcal{H}^2\)-almost everywhere the approximate tangent plane \(T_xC\) is \(J\) invariant and positively oriented. Then \(C\) is, except possibly at isolated points, the current of integration along a smooth two-dimensional Legendrian curve.