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Following earlier work of Loftin-McIntosh, we study minimal Lagrangianimmersions of the universal cover of a closed surface (of genus at least 2)into CH2, with prescribed data of a conformal structure plus a holomorphiccubic differential. We show existence and non-uniqueness of such minimalLagrangian immersions. We also establish the surface area with respect to theinduced metric as a Weil-Petersson potential function for the space ofholomorphic cubic differentials on the Riemann surface.

Holomorphic Cubic Differentials and Minimal Lagrangian Surfaces in CH2