Biharmonic Conformal Surfaces and Dirac Factorization I: Exact Spinorial Encoding and Scalar--Chiral Rigidity

Abstract

For maps from surfaces, harmonicity is conformally invariant and a conformal immersion is harmonic precisely when its image is minimal. Biharmonicity is a fourth-order extension of this theory, but it is not conformally invariant. A nonminimal immersion may therefore become biharmonic after a suitable change of the domain metric. In a three-dimensional space form, Ou's formulation reduces this problem to two coupled equations for the weighted mean curvature $U=λ^2H$: one scalar equation and one tangential equation. We ask whether these two equations can be organized as a single Dirac-type equation and what geometry is compatible with a first-order factorization. Restricting an ambient Killing spinor to the surface, we construct a natural Laplace-type operator $\mathscr B_c$ and prove that $\mathscr B_c(Uψ)=0$ is exactly equivalent to Ou's system. We then classify every factorization of $\mathscr B_c$ in the monic scalar--chiral class $(D+a+bω)(D+p+qω)$. In nonzero curvature, every such factorization is automatically mean-curvature-normalized and exists locally if and only if the surface has locally constant principal curvatures. The same rigidity holds for the normalized Euclidean branch; the remaining Euclidean factors form an exceptional holomorphic--antiholomorphic family characterized, away from planar points, by harmonicity of $\log(|A|^2-H^2)$. The rigidity mechanism is governed by a Dirac discriminant $9c-4|A|^2$. Its hypothetical non-CMC real branch reduces to a spherical Gauss--Codazzi system whose exact Frobenius torsion is strictly negative. Model examples finally show that factorization of the geometric operator is distinct from the existence of a positive conformal mode. Thus the scalar--chiral channel is complete but too rigid to generate new non-CMC examples, providing a precise baseline for broader Clifford-valued constructions.