Classification of biharmonic Lagrangian surfaces under intrinsic curvature conditions

Characterize biharmonic Lagrangian surfaces in complex space forms under intrinsic curvature conditions beyond the constant-Gaussian-curvature case, thereby resolving the broader classification problem that remains largely open.

Background

The paper emphasizes the difficulty of imposing intrinsic curvature constraints on biharmonic Lagrangian surfaces because biharmonicity is expressed through extrinsic quantities, including the mean curvature vector and the second fundamental form.

The authors resolve the specific case of proper biharmonic Lagrangian surfaces with constant Gaussian curvature in complex dimension two, proving that such surfaces have constant mean curvature and obtaining a complete classification. The broader classification under intrinsic curvature conditions is explicitly described as still incomplete.

References

Despite these advances, the classification of biharmonic Lagrangian surfaces under intrinsic curvature conditions remains largely open.

— Biharmonic Lagrangian surfaces with constant Gaussian curvature in $2$-dimensional complex space forms  (2610.05703 - Fu et al., 5 Oct 2026) in Section 1, Introduction