Stabilities of affine Legendrian submanifolds and their moduli spaces

We introduce the notion of affine Legendrian submanifolds in Sasakian manifolds and define a canonical volume called the $\phi$-volume as odd dimensional analogues of affine Lagrangian (totally real or purely real) geometry. Then we derive the second variation formula of the $\phi$-volume to obtain the stability result in some $\eta$-Einstein Sasakian manifolds. It also implies the convexity of the $\phi$-volume functional on the space of affine Legendrian submanifolds. Next, we introduce the notion of special affine Legendrian submanifolds in Sasaki-Einstein manifolds as a generalization of that of special Legendrian submanifolds. Then we show that the moduli space of compact connected special affine Legendrian submanifolds is a smooth Fr\'{e}chet manifold.