Minimal Lagrangian submanifolds of a Kähler manifold represent a very interesting class of submanifolds as they are Lagrangian with respect to the symplectic structure of the ambient space, while minimal with respect to the Riemannian structure. In this paper, we study minimal Lagrangian immersions of the universal cover of closed surfaces (of genus g ≥ 2) in CH2, with prescribed data (σ, tq), where σ is a conformal structure on the surface S, and qdz3 is a holomorphic cubic differential on the Riemann surface (S, σ). We show existence and non-uniqueness of such minimal Lagrangian immersions. We analyze the asymptotic behaviors for such immersions, and establish the surface area with respect to the induced metric as a Weil-Petersson potential function for the space of holomorphic cubic differentials on (S, σ).
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