# Persistence of boundary punctured holomorphic curves

Given a Legendrian $\Lambda$ in a contact manifold $(Y,\alpha)$, suppose one has a rigid J-holomorphic curve into the symplectization $(M,\omega)=(Y\times\mathbb{R},\mathrm{d}(e^t\alpha))$. As considered in Legendrian contact homology, the domain of the curve should be the boundary punctured unit disc, at boundary punctures it should converge to connecting Reeb chords and the boundary should be mapped to the Lagrangian $\Lambda\times\mathbb{R}$. Does this curve persist to exist under small deformations of the Legendrian or the Reeb flow?

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