Minimal Lagrangian surfaces in $${\mathbb {CH}^2}$$ and representations of surface groups into SU(2, 1)

Loftin, John; McIntosh, Ian

doi:10.1007/s10711-012-9717-1

Cited by 19 publications

(47 citation statements)

References 35 publications

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“…It is clear now that we have the following, see [28]: Lemma 2.1. The coordinate frame F CH 2 of a Lagrangian immersion into CH 2 is a smooth map F CH 2 : D → U 2,1 .…”

Section: 2mentioning

confidence: 90%

“…In this section, we discuss a loop group formulation of minimal Lagrangian surfaces in the complex hyperbolic plane CH 2 . Most of what we present can be found in [28]. We will use complex parameters and restrict generally to surfaces defined on some open and simply connected domain D of the complex plane C.…”

Section: Minimal Lagrangian Surfaces In Chmentioning

confidence: 99%

“…and defines a holomorphic potential η for f by the formula η = C −1 dC. 28 The potential η takes the form…”

Section: Flat Connections and Primitive Framesmentioning

confidence: 99%

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Survey on real forms of the complex A2(2)-Toda equation and surface theory

et al. 2019

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The classical result of describing harmonic maps from surfaces into symmetric spaces of reductive Lie groups [9] states that the Maurer-Cartan form with an additional parameter, the so-called loop parameter, is integrable for all values of the loop parameter. As a matter of fact, the same result holds for k-symmetric spaces over reductive Lie groups, [8].In this survey we will show that to each of the five different types of real forms for a loop group of A (2) 2 there exists a surface class, for which some frame is integrable for all values of the loop parameter if and only if it belongs to one of the surface classes, that is, minimal Lagrangian surfaces in CP 2 , minimal Lagrangian surfaces in CH 2 , timelike minimal Lagrangian surfaces in CH 2 1 , proper definite affine spheres in R 3 and proper indefinite affine spheres in R 3 , respectively.

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“…It is clear now that we have the following, see [28]: Lemma 2.1. The coordinate frame F CH 2 of a Lagrangian immersion into CH 2 is a smooth map F CH 2 : D → U 2,1 .…”

Section: 2mentioning

confidence: 90%

Section: Minimal Lagrangian Surfaces In Chmentioning

confidence: 99%

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Survey on real forms of the complex A2(2)-Toda equation and surface theory

et al. 2019

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“…Evaluating the matrix equation (1) above further, we obtain from the matrix entry (13) the equation ωz 2 = u 11 . Now the equations for the matrix entries (12) and (21) imply ∂zφ = 2ū 11 φ and that ψ is holomorphic respectively. As a result, φ = 0 and the 1-form α is exactly the Maurer-Cartan form of the SU 3 -frame of a minimal Lagrangian immersion, or ω, φ and ψ are all constant with ψ is non-vanishing, which gives a Lagrangian homogeneous surface.…”

Section: 5mentioning

confidence: 99%

“…We start by considering spaces F L j , j = 1, 2, 3 similar to [12]. Thus we obtain three 6-symmetric spaces of dimension 7 which all are actually equivariantly isomorphic to SU 3 /U 1 (Theorem 3.3 and Corollary 3.4).…”

Section: Introductionmentioning

confidence: 99%

Ruh–Vilms theorems for minimal surfaces without complex points and minimal Lagrangian surfaces in $$\mathbb {C}P^2$$

Dorfmeister

Kobayashi

2020

Math. Z.

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In this paper we investigate surfaces in CP 2 without complex points and characterize the minimal surfaces without complex points and the minimal Lagrangian surfaces by Ruh-Vilms type theorems. We also discuss the liftability of an immersion from a surface to CP 2 into S 5 in Appendix A.(2) 2 type [5]. While σ arose naturally in classical geometric investigations, the question arose, whether also σ 2 and σ 3 have a simple geometric meaning.The starting point for an approach to this question was the paper [13], which investigated arbitrary immersions from Riemann surfaces to CP 2 without complex points. However, since immersions from S 2 to CP 2 have been investigated intensively, for the final goals of this paper we exclude the Riemann surface S 2 from our discussion. More precisely, we consider an immersion f : M → CP 2 without complex points, where M is a Riemann surface different from S 2 . For our approach it is crucial to lift f to a map f : M → S 5 such that f = π • f where π : S 5 → CP 2 denotes the Hopf fibration. To clarify, when such a lift exists we have proven in the appendix that for a non-compact Riemann surface such a lift always exists and that in the case of a compact Riemann surface either the given immersion already has a global lift to S 5 or one can find a threefold covering τ :M → M of M such that the immersionf = f • τ :M → CP 2 admits a global lift to S 5 .So for this paper we always assume that any immersion under consideration does have a global lift to S 5 . For a more detailed investigation of liftable immersions f : M → CP 2 with global lift f : M → S 5 , we consider, to begin with, their composition with the universal coveringπ : D → M of M . In other word, we first investigate the case, where M = D is simply-connected.In this setting the ideas presented in [13] is applied. However, while in loc.cit. the investigation quickly moved on to consider minimal Lagrangian tori in CP 2 , in the present paper we consider a natural SU 3 -frame F(f) and thus obtain a setting similar to the one used in [7].

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Timelike minimal Lagrangian surfaces in the indefinite complex hyperbolic two-space

Dorfmesiter

Kobayashi

2020

Annali di Matematica

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It has been known for some time that there exist 5 essentially different real forms of the complex affine Kac-Moody algebra of type A(2) 2 and that one can associate 4 of these real forms with certain classes of "integrable surfaces", such as minimal Lagrangian surfaces in CP 2 and CH 2 , as well as definite and indefinite affine spheres in R 3 .In this paper we consider the class of timelike minimal Lagrangian surfaces in the indefinite complex hyperbolic two-space CH 2 1 . We show that this class of surfaces corresponds to the fifth real form.Moreover, for each timelike Lagrangian surface in CH 2 1 we define natural Gauss maps into certain homogeneous spaces and prove a Ruh-Vilms type theorem, characterizing timelike minimal Lagrangian surfaces among all timelike Lagrangian surfaces in terms of the harmonicity of these Gauss maps.

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Minimal Lagrangian surfaces in $${\mathbb {CH}^2}$$ and representations of surface groups into SU(2, 1)

Cited by 19 publications

References 35 publications

Survey on real forms of the complex A2(2)-Toda equation and surface theory

Survey on real forms of the complex A2(2)-Toda equation and surface theory

Ruh–Vilms theorems for minimal surfaces without complex points and minimal Lagrangian surfaces in $$\mathbb {C}P^2$$

Timelike minimal Lagrangian surfaces in the indefinite complex hyperbolic two-space

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