$\boldsymbol{\text{R}}^n$上某些多值调和函数的刻画
Characterisation of some multivalued harmonic functions on $\mathbb{R}^{n}$
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中文总结 AI 辅助
针对Rⁿ上的Z₂-调和函数，本文结合Donaldson与Dashen Yan的构造结果，证明其无穷远二次增长、光滑余维2分支集附近O(r^(3/2))增长的性质可唯一刻画该类函数，仅差Rⁿ的刚体运动。
中文摘要 AI 辅助
Simon Donaldson与Dashen Yan近期分别采用不同方法，在$\boldsymbol{\text{R}}^3$及$n\boldsymbol{\text{≥}}3$的$\boldsymbol{\text{R}}^n$上构造了$\boldsymbol{\text{Z}}_2$-调和函数；这类函数在无穷远处具二次增长，在光滑余维2分支集$\boldsymbol{\text{Σ}}$附近具$\boldsymbol{\text{O}}(r^{3/2})$的局部增长，本文证明该类性质可唯一刻画此类$\boldsymbol{\text{Z}}_2$-调和函数，仅差$\boldsymbol{\text{R}}^n$的刚体运动。
英文摘要
Using different approaches, Simon Donaldson and Dashen Yan recently constructed $\mathbb Z_2$-harmonic functions on $\mathbb R^3$ and on $\mathbb R^n$ for $n\geqslant 3$, respectively. Their examples have quadratic growth at infinity and $\mathcal{O}(r^{3/2})$ local growth near a smooth codimension-two branching set $Σ$. We show that these properties uniquely characterise such $\mathbb Z_2$-harmonic functions up to rigid motions of $\mathbb R^n$.
