Biharmonic Lagrangian surfaces with constant Gaussian curvature in $2$-dimensional complex space forms

Constant curvature is one of the most fundamental intrinsic conditions in submanifold geometry. Investigating the existence and classification of biharmonic submanifolds under intrinsic curvature constraints is generally difficult, since the biharmonic equation is expressed in terms of extrinsic data. In this paper, we obtain a complete classification of biharmonic Lagrangian surfaces with constant Gaussian curvature in 2-dimensional complex space forms. As a consequence, we classify the biharmonic Lagrangian surfaces with flat normal bundle and we prove that pseudo-umbilical biharmonic Lagrangian surfaces are necessarily minimal.