In this paper, for a compact special Legendrian submanifold with smooth boundary of contact Calabi--Yau manifolds we study the deformation of it with boundary confined in an appropriately chosen contact submanifold of codimension two which we also call a scafford (Definition 2.3) by analogy with Butsher . Our first result shows that it cannot be deformed, and the second claims that deformations of such a special Legendrian submanifold forms a one-dimensional smooth manifold under suitably weaker boundary confinement conditions. They may be viewed as supplements of the closed case considered by Tomassini and Vezzoni .
"Deformations of special Legendrian submanifolds with boundary." Osaka J. Math. 51 (3) 673 - 695, July 2014.