Minimal Lagrangian surfaces in $\Bbb S\sp 2 x $\Bbb S\sp 2$

Abstract

We deal with the minimal Lagrangian surfaces of the EinsteinKähler surface S 2 × S 2 , studying local geometric properties and showing that they can be locally described as Gauss maps of minimal surfaces in S 3 ⊂ R 4 . We also discuss the second variation of the area and characterize the most relevant examples by their stability behaviour.

Surfaces with parallel mean curvature vector in $\mathbb{S}^{2}×\mathbb{S}^{2}$ and $\mathbb{H}^{2}×\mathbb{H}^{2}$

“…Remark 3. This result is a generalization of Theorem 1 in [4], where the authors proved the result when = 1, i.e. when the ambient space is S 2 ×S 2 and the surface is compact.…”

On Hamiltonian stationary Lagrangian spheres in non-Einstein Kähler surfaces

“…Let γ be the genus of Σ. Following the same reasoning that in the proof of Proposition 3 in [4] (that is, computing the self-intersection number of Φ), as Φ is an embedding, we can prove that the degrees of φ and ψ satisfy γ = 1 + degφ degψ.…”

“…Using again (2.3), the assumption about the mean curvature joint with c 1 = c 2 say that the associated Jacobian C ≡ 0. But it is not difficult to check (see Lemma 2.1 in [4]) that if {e 1 , e 2 } is an oriented orthonormal basis of T p Σ, then…”

Lagrangian immersions in the product of Lorentzian two manifolds

“…Let Φ = (φ, ψ) : S → dS 2 × dS 2 be a Ω + -Lagrangian immersion with non null parallel mean curvature vector H. Following similar arguments with Theorem 1 of [6], consider an orthonormal frame (e 1 , e 2 ) with respect to the induced metric such that…”

“…where C := λ 2 µ 1 − λ 1 µ 2 =λ 1μ2 −λ 2μ1 and is called the associated Jacobian of the Lagrangian immersion Φ (see [6] and [13] for definitions and further details). Note that the vanishing of the associated Jacobian is equivalent with the fact that the Lagrangian immersion Φ is of projected rank one.…”