Tohoku Mathematical Journal

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

Tor Kajigaya

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

Article information

Tohoku Math. J. (2), Volume 65, Number 4 (2013), 523-543.

First available in Project Euclid: 6 December 2013

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Zentralblatt MATH identifier

Primary: 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42]
Secondary: 53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.)

Legendrian submanifold Sasakian manifold Legendrian stability


Kajigaya, Tor. Second variation formula and the stability of Legendrian minimal submanifolds in Sasakian manifolds. Tohoku Math. J. (2) 65 (2013), no. 4, 523--543. doi:10.2748/tmj/1386354294.

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