AI 中文总结
本研究证明了$\mathbb{C}^2$中多重调和一一映射的实雅可比行列式非零的四维Lewy型定理，还给出了调和拟共形同胚为双Lipschitz映射的判定准则。
AI 中文摘要
Lewy定理指出，平面区域之间的一一调和映射具有非零雅可比行列式。在维度至少为3的情况下，这一结论对一般的调和同胚并不成立。我们在附加多重调和性的复解析假设下，证明了一个四维版本的Lewy定理。更准确地说，若$F:\Omega\subset \mathbb C^2\to \mathbb C^2$是一个$C^2$类多重调和映射，且在点$p$的邻域内是一一映射，则$F$的实雅可比行列式在$p$点处非零。\n 该证明是局部性的。若雅可比行列式为零，则$F$的一个非平凡实线性投影将是满足$df(p)=0$的全纯函数$f$的实部。因此，该投影的水平超曲面将是形如$\{\operatorname{Re} f=0\}$的实解析芽。我们证明了这类芽在$f$的临界点处不可能是局部平坦的。其障碍是拓扑层面的：局部平坦性要求超曲面芽的局部同调与实超平面的局部同调一致，进而要求所有足够小的容许链环具有$S^2$的整同调，尤其是欧拉示性数。在约化情形下，平面曲线奇点$f^{-1}(0)$的Milnor开书会导出矛盾的欧拉示性数公式；而在非约化情形下，局部正规形会产生两个以上的局部互补分支。\n 随后我们证明了关于调和拟共形映射的一个独立的双Lipschitz准则：从单位球到有界$C^1$-Dini区域的调和拟共形同胚，若它本身已是局部$C^1$微分同胚，则它是双Lipschitz的。
英文摘要
Lewy's theorem says that a one-to-one harmonic mapping between plane domains has nonvanishing Jacobian. In dimensions at least three this statement is false for general harmonic homeomorphisms. We prove a four-dimensional Lewy theorem under the additional complex-analytic assumption of pluriharmonicity. More precisely, if \[ F:Ω\subset \mathbb C^2\to \mathbb C^2 \] is a \(C^2\) pluriharmonic mapping which is one-to-one in a neighborhood of a point \(p\), then the real Jacobian of \(F\) is nonzero at \(p\). The proof is local. If the Jacobian vanished, a nontrivial real linear projection of \(F\) would be the real part of a holomorphic function \(f\) with \(df(p)=0\). The level hypersurface of this projection would therefore be a real analytic germ of the form \[ \{\operatorname{Re} f=0\}. \] We prove that such a germ cannot be locally flat at a critical point of \(f\). The obstruction is topological: local flatness forces the local homology of the hypersurface germ to agree with that of a real hyperplane, and hence forces every sufficiently small admissible link to have the integral homology, in particular the Euler characteristic, of \(S^2\). In the reduced case the Milnor open book of the plane curve singularity \(f^{-1}(0)\) gives a contradictory Euler-characteristic formula, while in the nonreduced case the local normal form produces more than two local complementary components. We then prove a separate bi-Lipschitz criterion for harmonic quasiconformal mappings: a harmonic quasiconformal homeomorphism from the unit ball onto a bounded \(C^1\)-Dini domain is bi-Lipschitz, provided it is already a local \(C^1\)-diffeomorphism.
Comments21 pages
