SECOND VARIATIONAL 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 $\boldsymbol{R}^{2n+1}(-3)$.