Minimal Lagrangian surfaces in the tangent bundle of a Riemannian surface

Anciaux, Henri; Guilfoyle, Brendan; Romon, Pascal

doi:10.1016/j.geomphys.2010.09.017

Cited by 10 publications

(15 citation statements)

References 6 publications

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“…This result is a generalization of previous work on the tangent bundle of a Riemannian surface (see [10], [11], [3]). The authors wish to thank Brendan Guilfoyle for his valuable suggestions and comments.…”

Section: Introductionsupporting

confidence: 73%

A canonical structure on the tangent bundle of a pseudo- or para-Kähler manifold

Anciaux

Romon²

2014

Monatsh Math

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It is a classical fact that the cotangent bundle T * M of a differentiable manifold M enjoys a canonical symplectic form Ω * . If (M, J, g, ω) is a pseudo-Kähler or para-Kähler 2n-dimensional manifold, we prove that the tangent bundle T M also enjoys a natural pseudo-Kähler or para-Kähler structure (J,g, Ω), where Ω is the pull-back by g of Ω * andg is a pseudo-Riemannian metric with neutral signature (2n, 2n). We investigate the curvature properties of the pair (J,g) and prove that:g is scalar-flat, is not Einstein unless g is flat, has nonpositive (resp. nonnegative) Ricci curvature if and only if g has nonpositive (resp. nonnegative) Ricci curvature as well, and is locally conformally flat if and only if n = 1 and g has constant curvature, or n > 2 and g is flat. We also check that (i) the holomorphic sectional curvature of (J,g) is not constant unless g is flat, and (ii) in n = 1 case, thatg is never anti-self-dual, unless conformally flat. MSC : 32Q15, 53D05

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Section: Introductionsupporting

confidence: 73%

A canonical structure on the tangent bundle of a pseudo- or para-Kähler manifold

Anciaux

Romon²

2014

Monatsh Math

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“…we find H μ 2 , and finally the mean curvature vector H of is: 2 , which means that the surface is area stationary iff…”

Section: Non-existence Of Rank One Area Stationary Surfacesmentioning

confidence: 99%

“…More recently, Anciaux, Guilfoyle and Romon [2] have studied Lagrangian area-stationary surfaces in TN, with N being an oriented Riemannian surface and the neutral Kähler structure generalising that of the space of oriented geodesics in Euclidean and Lorentzian 3-space.…”

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confidence: 99%

On area stationary surfaces in the space of oriented geodesics of hyperbolic 3-space

Georgiou¹

2012

MATH. SCAND.

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We study area-stationary surfaces in the space L(H 3 ) of oriented geodesics of hyperbolic 3-space, endowed with the canonical neutral Kähler structure. We prove that every holomorphic curve in L(H 3 ) is an area-stationary surface. We then classify Lagrangian area-stationary surfaces in L(H 3 ) and prove that the family of parallel surfaces in H 3 orthogonal to the geodesics γ ∈ form a family of equidistant tubes around a geodesic. Finally we find an example of a two parameter family of rotationally symmetric area-stationary surfaces that are neither Lagrangian nor holomorphic.The last two decades has seen increasing interest in spaces L(M) of oriented geodesics of a manifold M, with particular attention to their rich geometric structure. In the case of the space L(E 3 ) of oriented affine lines in Euclidean 3-space this interest can be traced back over a hundred years to Weierstrass's construction of minimal surfaces [16] and Whittaker's solutions to the wave equation [17].A natural complex structure on L(E 3 ) was considered by Hitchin to construct monopoles in E 3 [11], and then Guilfoyle and Klingenberg understood that the canonical symplectic structure on L(E 3 ) is compatible with this complex structure [7], [8] and that the associated Kähler metric is of neutral signature. Salvai subsequently proved that this neutral Kähler metric is (up to addition of the round metric) the unique metric on L(E 3 ) that is invariant under Euclidean motions [12]. This Kähler structure has recently been used by Guilfoyle and Klingenberg to solve an 80 year old conjecture of Carathéodory [10].More recently, Anciaux, Guilfoyle and Romon [2] have studied Lagrangian area-stationary surfaces in TN, with N being an oriented Riemannian surface and the neutral Kähler structure generalising that of the space of oriented geodesics in Euclidean and Lorentzian 3-space.In addition, Salvai constructed a neutral Kähler metric on the space L(H 3 ) of oriented geodesics in hyperbolic 3-space [13], while the geometry of L(H 3 )

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“…where H, K denote the mean and the Gauss curvature of S, respectively [6]. The neutral Kähler structures on the space of oriented great circles in the three sphere S 3 and the space of oriented space-like geodesics in the anti De Sitter 3space AdS 3 can both be identified with the product structures, L(S 3 ) = S 2 × S 2 and L + (AdS 3 ) = H 2 × H 2 .…”

Section: Introductionmentioning

confidence: 99%

“…Here a generic property is one that holds almost everywhere. Note that Theorem 3 is no longer true when (Σ 1 , g 1 ) and (Σ 2 , g 2 ) are both flat, since there exist projected rank two minimal Lagrangian immersions in the complex Euclidean space C 2 endowed with the pseudo-Hermitian product structure [6]. Minimality is the first order condition for a submanifold to be volume-extremizing in its homology class.…”

Section: Introductionmentioning

confidence: 99%

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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Minimal Lagrangian surfaces in the tangent bundle of a Riemannian surface

Cited by 10 publications

References 6 publications

A canonical structure on the tangent bundle of a pseudo- or para-Kähler manifold

A canonical structure on the tangent bundle of a pseudo- or para-Kähler manifold

On area stationary surfaces in the space of oriented geodesics of hyperbolic 3-space

On minimal Lagrangian surfaces in the product of Riemannian two manifolds

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