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 ϕ-volume as odd dimensional analogues of affine Lagrangian (totally real or purely real) geometry. Then we derive the second variation formula of the ϕ-volume to obtain the stability result in some η-Einstein Sasakian manifolds. It also implies the convexity of the ϕ-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 Fréchet manifold.