AI 中文总结
本文研究欧氏空间上分支轨迹为椭球面、平面二次曲线的非退化$\boldsymbol{\rm Z}_2$调和函数，证明相关参数映射性质、构造特定分支轨迹的调和函数并揭示其与特殊拉格朗日三维流形的关联。
AI 中文摘要
我们研究欧氏空间上分支轨迹为椭球面与平面二次曲线的非退化$\boldsymbol{\rm Z}_2$调和函数。首先证明与Yan椭球族相关的参数映射是单射；结合Yan的满射结果，这确定了椭球模型（不计欧氏运动与整体符号）对应指标为$n-1$且临界值为正的非退化调和二次多项式。在$\boldsymbol{\rm R}^3$中，我们利用修正椭球坐标构造平面双曲线邻域内的非退化$\boldsymbol{\rm Z}_2$调和函数，随后证明以平面双曲线为分支轨迹的全局临界$\boldsymbol{\rm Z}_2$调和函数在无穷远处不存在有限Almgren频率。利用修正抛物坐标，我们构造以任意给定平面抛物线为分支轨迹的全局非退化$\boldsymbol{\rm Z}_2$调和函数，得到精确渐近展开与缩尺极限。最后，我们将抛物模型识别为Joyce构造的一类特殊拉格朗日三维流形的无穷小三值图势。
英文摘要
We study nondegenerate $\mathbb Z_2$ harmonic functions on Euclidean spaces with branching loci given by ellipsoids and planar conics. We first prove that the parameter map associated with Yan's ellipsoidal family is injective. Together with Yan's surjectivity result, this identifies the ellipsoidal models, up to Euclidean motions and overall sign, with nondegenerate harmonic quadratic polynomials of index \(n-1\) and positive critical value. In \(\mathbb R^3\), we use modified ellipsoidal coordinates to construct a nondegenerate $\mathbb Z_2$ harmonic function in a neighborhood of any planar hyperbola. We then show that no global critical $\mathbb Z_2$ harmonic function with a planar hyperbola as branch locus can have finite Almgren frequency at infinity. Using modified paraboloidal coordinates, we construct a global nondegenerate $\mathbb Z_2$ harmonic function with any prescribed planar parabola as branch locus and obtain a precise asymptotic expansion and scale-down limit. Finally, we identify the parabolic model as the infinitesimal two-valued graph potential of a family of special Lagrangian three-folds constructed by Joyce.
