双调和共形曲面与Dirac分解I：精确旋量编码与标量-手性刚性
Biharmonic Conformal Surfaces and Dirac Factorization I: Exact Spinorial Encoding and Scalar--Chiral Rigidity
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- Southern Methodist University（南方卫理公会大学）
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中文总结 AI 辅助
本文研究双调和共形曲面，通过构造Laplace型算子将Ou方程组等价为单个Dirac方程，分类标量-手性分解，证明其刚性由Dirac判别式控制，为更广构造提供基线。
中文摘要 AI 辅助
对于从曲面出发的映射，调和性在共形变换下保持不变，且一个共形浸入是调和的当且仅当其像为极小曲面。双调和性是该理论的四阶推广，但它不是共形不变的。因此，一个非极小的浸入在适当改变定义域度量后可能变为双调和的。在三维空间形式中，Ou的表述将该问题简化为关于加权平均曲率$U=\lambda^2H$的两个耦合方程：一个标量方程和一个切向方程。我们问这两个方程能否组织成单个Dirac型方程，以及何种几何与一阶分解相容。将环境Killing旋量限制到曲面上，我们构造了一个自然的Laplace型算子$\mathscr B_c$，并证明$\mathscr B_c(U\psi)=0$恰好等价于Ou的系统。然后我们在首一标量-手性类$(D+a+b\omega)(D+p+q\omega)$中分类$\mathscr B_c$的每一个分解。在非零曲率下，每个这样的分解自动是平均曲率归一化的，并且局部存在当且仅当曲面具有局部常主曲率。相同的刚性对归一化欧几里得分支成立；其余欧几里得因子构成一个例外全纯-反全纯族，其特征为在非平面点之外由$\log(|A|^2-H^2)$的调和性刻画。刚性机制由Dirac判别式$9c-4|A|^2$控制。其假设的非CMC实分支归结为一个球面Gauss-Codazzi系统，其精确Frobenius挠率严格为负。模型例子最终表明几何算子的分解与正共形模式的存在性是不同的。因此标量-手性通道是完备的但过于刚性，无法产生新的非CMC例子，为更广泛的Clifford值构造提供了精确的基线。
英文摘要
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.
