Tohoku Mathematical Journal
2013
December
SECOND SERIES VOL. 65, NO. 4

Tohoku Math. J.
65 (2013), 523-543

Title SECOND VARIATION FORMULA AND THE STABILITY OF LEGENDRIAN MINIMAL SUBMANIFOLDS IN SASAKIAN MANIFOLDS

Author Toru Kajigaya

(Received August 22, 2012, revised February 6, 2013)
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 ${\boldsymbol R}^{2n+1}(-3)$.

Mathematics Subject Classification. Primary 53C42; Secondary, 53C25.
Key words and phrases. Legendrian submanifold, Sasakian manifold, Legendrian stability.

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