AI 中文总结
研究三维保向调和拟正则映射，用齐次爆破论证证明其分支集为空，给出Lewy定理的三维有界畸变形式，还证明相关拓扑障碍及仿射刘维尔定理，构造反例说明齐次障碍的严格性并对高维等变调和三次模型分类。
AI 中文摘要
我们证明了一个非常数的保向调和拟正则映射\(f:\Omega\subset\mathbb{R}^3\to\mathbb{R}^3\)的分支集为空。等价地，这样的映射处处有\(J_f>0\)且局部是实解析微分同胚。这给出了Lewy定理的三维有界畸变形式。证明基于齐次爆破论证。我们还证明了一个伴随的拓扑障碍，导出了\(\mathbb{R}^3\)中整体调和拟正则映射的仿射刘维尔定理。最后，我们对高维\(O(n - 1)\)等变调和三次模型进行了分类。
英文摘要
Lewy's classical theorem asserts that a one-to-one planar harmonic mapping has nonvanishing Jacobian. We prove a three-dimensional bounded-distortion analogue: if \[ f:Ω\subset \mathbb R^3\to \mathbb R^3 \] is nonconstant, sense-preserving, quasiregular, and harmonic componentwise, then \(J_f>0\) throughout \(Ω\). Thus harmonic quasiconformal mappings between domains in three-space are local harmonic diffeomorphisms. The new point is the Lewy-type differential conclusion \(J_f\neq0\), not merely topological local invertibility, which is already known for sufficiently smooth quasiregular mappings. The proof is by blow-up. A hypothetical zero of \(J_f\) produces a nonconstant homogeneous harmonic polynomial quasiregular mapping \(P:\mathbb R^3\to\mathbb R^3\) of degree \(m>1\). We exclude such homogeneous blow-ups by a second-order trace identity for \(J_P|_{S^2}\): after normalizing the first jet at a positive minimum, the identity gives a negative spherical trace, contradicting the maximum principle. We also derive an affine Liouville theorem for entire harmonic quasiregular mappings in \(\mathbb R^3\).
Comments23 pages
