Finite-gap minimal Lagrangian surfaces in $CP^2$

In this paper we suggest a method for constructing minimal Lagrangian immersions of $R^2$ in $CP^2$ with induced diagonal metric in terms of Baker-Akhiezer functions of algebraic curves.

💡 Research Summary

The paper presents a systematic construction of minimal Lagrangian immersions of the Euclidean plane into the complex projective space CP² by exploiting the finite‑gap integration technique. The authors start from the geometric requirements: a Lagrangian immersion must annihilate the ambient Kähler form ω (i.e., ω|_Σ = 0), and minimality demands vanishing mean curvature, which in the language of harmonic maps translates into the immersion being a conformal harmonic map. Directly solving the resulting nonlinear PDE system is notoriously difficult, so the authors turn to the theory of integrable systems, where a large class of special solutions—finite‑gap or algebro‑geometric solutions—can be written explicitly in terms of data on a compact Riemann surface (the spectral curve) and a Baker‑Akhiezer function defined on it.

The construction proceeds as follows. Choose a smooth algebraic curve Γ of genus g together with three marked points P₁, P₂ and P∞. Local parameters k₁⁻¹, k₂⁻¹, k∞⁻¹ are introduced near these points. A divisor D of degree g+1 is fixed to prescribe the pole structure of the Baker‑Akhiezer function ψ(p; x, y). The function ψ is uniquely characterized by three properties: (i) exponential asymptotics ψ ∼ exp(k₁x) (resp. exp(k₂y)) as p → P₁ (resp. P₂); (ii) meromorphicity on Γ with only simple poles at the points of D; (iii) a reality condition enforced by an anti‑holomorphic involution σ on Γ, guaranteeing that ψ(p̄) = \overline{ψ(σ(p))}. These conditions guarantee that ψ depends analytically on the planar variables (x, y) and that its three components ψ₁, ψ₂, ψ₃ can be assembled into a homogeneous coordinate map \