On Hamiltonian stationary Lagrangian spheres in non-Einstein Kähler surfaces

Castro, Ildefonso; Torralbo, Francisco; Urbano, Francisco J.

doi:10.1007/s00209-011-0862-2

Cited by 2 publications

(2 citation statements)

References 14 publications

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“…Moreover, the assumption that Φ is of projected rank two, implies that λ 1 µ 2 −λ 2 µ 1 = 0 for every p ∈ S. If H ǫ is the mean curvature vector of the immersion Φ, consider the one form a H ǫ defined by a H ǫ = G ǫ (JH ǫ , ·). Since Φ is Lagrangian, it is known from [9] that…”

Section: Projected Rank Two Lagrangian Surfacesmentioning

confidence: 99%

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On minimal Lagrangian surfaces in the product of Riemannian two manifolds

Georgiou¹

2015

Tohoku Math. J. (2)

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Let (Σ 1 , g 1 ) and (Σ 2 , g 2 ) be connected, complete and orientable Riemannian two manifolds. Consider the two canonical Kähler structures (and J is the canonical product complex structure. Thus for ǫ = 1 the Kähler metric G + is Riemannian while for ǫ = −1, G − is of neutral signature. We show that the metric G ǫ is locally conformally flat iff the Gauss curvatures κ(g 1 ) and κ(g 2 ) are both constants satisfying κ(g 1 ) = −ǫκ(g 2 ).We also give conditions on the Gauss curvatures for which every G ǫ -minimal Lagrangian surface is the product γ 1 × γ 2 ⊂ Σ 1 × Σ 2 , where γ 1 and γ 2 are geodesics of (Σ 1 , g 1 ) and (Σ 2 , g 2 ), respectively. Finally, we explore the Hamiltonian stability of projected rank one Hamiltonian G ǫ -minimal surfaces.

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Section: Projected Rank Two Lagrangian Surfacesmentioning

confidence: 99%

“…If H ǫ is the mean curvature vector of the immersion Φ, consider the one form a H ǫ defined by a H ǫ = G ǫ (JH ǫ , •). Since Φ is Lagrangian, it is known from [9] that…”

Section: Projected Rank Two Lagrangian Surfacesmentioning

confidence: 99%

On minimal Lagrangian surfaces in the product of Riemannian two manifolds

Georgiou¹

2015

Tohoku Math. J. (2)

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Lagrangian submanifolds in the homogeneous nearly Kähler $${\mathbb {S}}^3 \times {\mathbb {S}}^3$$ S 3 × S 3

2017

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In this paper, we investigate Lagrangian submanifolds in the nearly Kähler S 3 × S 3 . We construct a new example which is a flat Lagrangian torus. We give a complete classification of all the Lagrangian immersions of spaces of constant sectional curvature in the nearly Kähler S 3 × S 3 . As a corollary, we obtain that the radius of a round Lagrangian sphere in the nearly Kähler S 3 × S 3 can only be 2.

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On Hamiltonian stationary Lagrangian spheres in non-Einstein Kähler surfaces

Cited by 2 publications

References 14 publications

On minimal Lagrangian surfaces in the product of Riemannian two manifolds

On minimal Lagrangian surfaces in the product of Riemannian two manifolds

Lagrangian submanifolds in the homogeneous nearly Kähler $${\mathbb {S}}^3 \times {\mathbb {S}}^3$$ S 3 × S 3

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