AI 中文总结
本研究分析无源带标架Beltrami-Vekua方程在变换下的模空间，给出无零点情形的双曲质量密度与相位曲率流分类及不变量范围，并将分类扩展至带零点场的相位可积扇区，构造了等质量等电荷的不等价方程实例。
AI 中文摘要
我们研究未知函数重组、缩放变换以及保定向变量替换下，无源带标架Beltrami-Vekua方程的模空间。在每个有界单连通域上，这类方程可约化为单位圆盘上的$w_{\bar z}=B\bar w$，其剩余对称恰好由无零点全纯规范变换与Möbius变换给出。当$B$无零点时，方程在模掉Möbius变换后由两个数据完全分类：双曲质量密度$\vartheta=\tfrac14(1-|z|^2)^2|B|^2$，以及相位曲率流$K=\Delta\arg B$，其在$C^2$正则性下的双曲密度为$\kappa=\Delta_{\mathrm{hyp}}\arg B$。我们确定了这些不变量的精确取值范围：所有正Hölder密度与所有作为Hölder相位的拉普拉斯算子产生的相位流均可实现。对于带零点的场，分类可通过三元组$(\vartheta,d\eta,d{\star}\eta)$扩展至相位可积扇区，其中$\eta=\operatorname{Im}(dB/B)$。在驯顺扇区中，$d\eta$是原子电荷测度，而$d{\star}\eta$承载剩余的相位曲率。因此，伪解析质量与电荷是更大的无限维模空间的数值投影。我们还构造了具有相等质量、相等电荷但不等价的方程实例。
英文摘要
We study the moduli of sourceless framed Beltrami-Vekua equations under recombinations of the unknown, scalings, and orientation-preserving changes of variables. On every bounded simply connected domain, such an equation reduces to $w_{\bar z}=B\bar w$ on the unit disk, with residual symmetries given exactly by zero-free holomorphic gauges and Möbius transformations. When $B$ is zero-free, the equation is completely classified by two data modulo Möbius: the hyperbolic mass density $\vartheta=\tfrac14(1-|z|^2)^2|B|^2$ and the phase-curvature current $K=Δ\arg B$, whose hyperbolic density at $C^2$ regularity is $κ=Δ_{\mathrm{hyp}}\arg B$. We determine the exact range of these invariants: every positive Hölder density and every phase current arising as the Laplacian of a Hölder phase occur. For fields with zeros, the classification extends to the phase-integrable sector through the triple $(\vartheta,dη,d{\star}η)$, where $η=\operatorname{Im}(dB/B)$. On the tame sector, $dη$ is the atomic charge measure, while $d{\star}η$ carries the remaining phase curvature. Thus the pseudo-analytic mass and charge are numerical projections of a larger infinite-dimensional moduli space. Explicit equal-mass, equal-charge, inequivalent equations are exhibited.
Comments30 pages. Comments are welcome
