全维Lewy定理：通过Milnor单值性研究多重调和映射
An all-dimensional Lewy theorem for pluriharmonic mappings via Milnor monodromy
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中文总结 AI 辅助
本文通过奇点理论中的Milnor单值性，证明了任意复维度下多重调和映射局部单射当且仅当实Jacobian非零，并应用于拟共形多重调和同胚的双Lipschitz性质。
中文摘要 AI 辅助
Hengartner提出了一个问题：在等维复欧几里得域之间的多重调和映射，是否恰好在其实Jacobian非零的点处局部一对一？Naser早先在复二维情形下声称了这一结论，而Kalaj最近的一个定理为该维度提供了完整证明。我们在所有复维度上证明了该结果。主要工具是一个具有独立兴趣的奇点理论刻画：若$q\colon(\C^n,0)\to(\C,0)$是一个非恒定的全纯芽，则$dq(0)\ne0$当且仅当（等价地，每一个）相位超曲面\\(\{\Repart(e^{-i\theta}q)=0\}\\)在原点是理性同调$(2n-1)$-流形。困难的方向结合了局部锥形结构、Alexander对偶、Milnor纤维化以及A'Campo关于局部单值性Lefschetz数的消失定理。该论证既不需要孤立临界点，也不需要约化性假设。由此得出，任何多重调和自维映射的实临界集与其局部非单射集完全一致。作为定量应用，将我们的定理与Kalaj的边界正则性定理相结合，表明在所有复维度中，从单位球到有界$C^1$-Dini域上的拟共形多重调和同胚是双Lipschitz的。
英文摘要
Hengartner asked whether a pluriharmonic mapping between equidimensional complex Euclidean domains is locally one-to-one precisely at the points where its real Jacobian is nonzero. Naser had earlier claimed the conclusion in complex dimension two, and a recent theorem of Kalaj supplies a complete proof in that dimension. We prove the result in every complex dimension. The main ingredient is a singularity-theoretic characterization which is of independent interest: if $q\colon(\C^n,0)\to(\C,0)$ is a nonconstant holomorphic germ, then $dq(0)\ne0$ if and only if one (equivalently, every) phase hypersurface \( \{\Repart(e^{-iθ}q)=0\} \) is a rational homology $(2n-1)$-manifold at the origin. The difficult direction combines local conical structure, Alexander duality, the Milnor fibration, and A'Campo's vanishing theorem for the Lefschetz number of local monodromy. The argument requires neither an isolated critical point nor a reducedness hypothesis. It follows that the real critical set of any pluriharmonic self-dimensional mapping agrees exactly with its local noninjectivity set. As a quantitative application, combining our theorem with Kalaj's boundary regularity theorem shows that, in every complex dimension, a quasiconformal pluriharmonic homeomorphism of the unit ball onto a bounded $C^1$-Dini domain is bi-Lipschitz.
发表机构
- Shenzhen Polytechnic University（深圳职业技术大学）
- Hunan First Normal University（湖南第一师范学院）
机构由 AI 辅助整理，请以论文原文为准。
