Second variation formula and the stability of Legendrian minimal submanifolds in Sasakian manifolds

Abstract

In this paper, we investigate compact Legendrian submanifolds $L$ in Sasakian manifolds $M$, which have extremal volume under Legendrian deformations. We call such a submanifold $L$-minimal Legendrian submanifold. We derive the second variational formula for the volume of $L$ under Legendrian deformations in $M$. Applying this formula, we investigate the stability of $L$-minimal Legendrian curves in Sasakian space forms, and show the $L$-instability of $L$-minimal Legendrian submanifolds in $S^{2n+1}(1)$. Moreover, we give a construction of $L$-minimal Legendrian submanifolds in ${\boldsymbol R}^{2n+1}(-3)$.