Abstract
Let (M,J,ω) be a compact toric Kähler manifold of dimℂ M=n and L a regular orbit of the T n-action on M. In the present paper, we investigate Hamiltonian stability of L, which was introduced by Y.-G. Oh (Invent. Math. 101, 501–519 (1990); Math. Z. 212, 175–192) (1993)). As a result, we prove any regular orbit is Hamiltonian stable when (M,ω)=ℂℙn,ωFS) and (M,ω)=ℂℙn 1× ℂℙn 2,aωFS⊕ bωFS), where ωFS is the Fubini–Study Kähler form and a and b are positive constants. Moreover, they are locally Hamiltonian volume minimizing Lagrangian submanifolds.
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Abreu, M.: Kähler geometry of toric manifolds in symplectic coordinates. In: Eliashberg, Y., Khesin, B., Lalonde, F. (eds.) Symplectic and Contact Topology: Interactions and Perspectives, pp. 1–24. AMS (2003)
Goldstein, E.: Calibrated fibrations on non-compact manifolds via group actions. Duke Math. J. 110(2), 309–344 (2001)
Guillemin, V.: Kaehler structures on toric varieties. J. Diff. Geom. 40, 285–309 (1994)
Iriyeh, H., Ono, H., Sakai, T.: Integral Geometry and Hamiltonian volume minimizing property of a totally geodesic Lagrangian torus in S2 , S2. Proc. Jpn. Acad. Ser. A 79(10), 167–170 (2003)
Oh, Y.-G.: Second variation and stabilities of minimal lagrangian submanifolds in Kähler manifolds. Invent. Math. 101, 501–519 (1990)
Oh, Y.-G.: Volume minimization of Lagrangian submanifolds under Hamiltonian deformations. Math. Z. 212, 175–192 (1993)
Ono, H.: Hamiltonian stability of Lagrangian tori in toric Kähler manifolds II, manuscript in preparation
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Mathematics subject classifications (2000): 53C42, 53D12
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Ono, H. Hamiltonian stability of Lagrangian tori in toric Kähler manifolds. Ann Glob Anal Geom 31, 329–343 (2007). https://doi.org/10.1007/s10455-006-9037-5
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DOI: https://doi.org/10.1007/s10455-006-9037-5