## Tohoku Mathematical Journal

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

Tor Kajigaya

#### 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

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

Digital Object Identifier
doi:10.2748/tmj/1386354294

Mathematical Reviews number (MathSciNet)
MR3161432

Zentralblatt MATH identifier
1287.53054

#### 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

#### References

• A. Amarzaya and Y. Ohnita, Hamiltonian stability of certain minimal Lagrangian submanifolds in complex projective spaces, Tohoku Math. J. 55 (2003), 583–610.
• C. Baikoussis and D. E. Blair, Finite type integral submanifolds of the contact manifold ${\boldsymbol R}^{2n+1}(-3)$, Bull. Inst. Math. Acad. Sinica 19 (1991), no. 4, 327–350.
• C. Baikoussis and D. E. Blair, On Legendre curves in contact 3–manifolds, Geom. Dedicata 49 (1994), 135–142.
• M. Belkhelfa, F. Dillen and J. Inoguchi, Parallel surfaces in the real special linear group $SL(2, {\boldsymbol R})$, Bull. Austral. Math. Soc. 65 (2002), 183–189.
• D. E. Blair, Riemannian geometry of contact and symplectic manifolds, Progress in Math. 203, Birkhäuser, Inc., Boston, MA, 2002.
• C. P. Boyer and K. Galicki, Sasakian geometry, Oxford Math. Monogr., Oxford University Press, Oxford, 2008.
• I. Castro, H. Li and F. Urbano, Hamiltonian-minimal Lagrangian submanifolds in complex space forms, Pacific J. Math. 227 (2006), no. 1, 43–63.
• H. Iriyeh, Hamiltonian minimal Lagrangian cones in ${\boldsymbol C}^m$, Tokyo J. Math. 28 (2005), no. 1, 91–107.
• H. B. Lawson, Jr. and J. Simons, On stable currents and their application to global problems in real and complex geometry, Ann. of Math. 98 (1973), 427–450.
• Y. G. Oh, Second variation and stability of minimal Lagrangian submanifolds, Invent. Math. 101 (1990), 501–519.
• Y. G. Oh, Volume minimization of Lagrangian submanifolds under Hamiltonian deformations, Math. Zeit. 212 (1993), 175–192.
• H. Ono, Second variation and Legendrian stabilities of minimal Legendrian submanifolds in Sasakian manifolds, Differential Geom. Appl. 22 (2005), 327–340.
• H. Reckziegel, A correspondence between horizontal submanifolds of Sasakian manifolds and totally real submanifolds of Kähler manifolds, Topics in differential geometry, Vol. I, II (Debrecen, 1984), 1063–1081, Colloq. Math. Soc. János Bolyai, 46, North-Holland, Amsterdam, 1988.
• R. Schoen and J. Wolfson, Minimizing area among Lagrangian surfaces: The mapping problem, J. Differential Geom. 58 (2001), 1–86.