Equivalent Characterisations of Singularity in Curvature-Coupled Membrane Functionals

We study a curvature-based functional describing the coupling between isotropic bending and curvature anisotropy in membrane shapes. The functional is reformulated in invariant geometric form, enabling a structural analysis from three complementary viewpoints: the quadratic curvature kernel, the linearised shape operator, and the Euler–Lagrange equations. In each formulation, a common threshold is identified at which the highest-order curvature contribution degenerates. This provides an equivalent characterisation of the resulting singular behaviour, linking it to a loss of ellipticity of the associated operator and to a vanishing of curvature stiffness in the deviatoric mode. The coincidence of these conditions across all three perspectives demonstrates that the singularity is intrinsic to the model and independent of the chosen representation. These results clarify the structural origin of degeneracy in curvature-coupled membrane theories.