Tohoku Mathematical Journal

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

Tor Kajigaya

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

Article information

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

Dates
First available in Project Euclid: 6 December 2013

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1386354294

Digital Object Identifier
doi:10.2748/tmj/1386354294

Mathematical Reviews number (MathSciNet)
MR3161432

Zentralblatt MATH identifier
1287.53054

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

Keywords
Legendrian submanifold Sasakian manifold Legendrian stability

Citation

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. https://projecteuclid.org/euclid.tmj/1386354294


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