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 n-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 Frechet manifold. (C) 2016 Elsevier B.V. All rights reserved.