论文标题
平面域上的谐波映射的单价和准文献扩展的标准
Criteria for univalence and quasiconformal extension for harmonic mappings on planar domains
论文作者
论文摘要
如果$ω$是$ \ operline {\ mathbb c} $中的简单连接的域，则根据Ahlfors-Gehring定理，$ω$是准确的，并且仅当存在足够的情况下，当时存在足够的条件，可以使holomorphic函数在$ω$中的生长中与其Schwarzian derivitivativitivativitivativitivativatived of to $ω$之间的关系。我们通过证明对准风险的单价标准将该定理扩展到谐波映射。我们还表明，满足该标准的映射承认同构扩展为$ \ operline {\ mathbb c} $，并且在$ω$中的准真实性的附加假设下，他们承认quasiconformalformal formenformal formenformal {\ + edline {\ mathbb c} $。 Osgood，Beardon和Gehring已将Ahlfors-Gehring定理扩展到有限连接的域$ω$，他们表明，当$ \partialΩ$的组件是点或quasicircles的分量时，Schwarzian的Univalentions in $ω$ in $ω$。我们将此定理推广到谐波映射。
If $Ω$ is a simply connected domain in $\overline{\mathbb C}$ then, according to the Ahlfors-Gehring theorem, $Ω$ is a quasidisk if and only if there exists a sufficient condition for the univalence of holomorphic functions in $Ω$ in relation to the growth of their Schwarzian derivative. We extend this theorem to harmonic mappings by proving a univalence criterion on quasidisks. We also show that the mappings satisfying this criterion admit a homeomorphic extension to $\overline{\mathbb C}$ and, under the additional assumption of quasiconformality in $Ω$, they admit a quasiconformal extension to $\overline{\mathbb C}$. The Ahlfors-Gehring theorem has been extended to finitely connected domains $Ω$ by Osgood, Beardon and Gehring, who showed that a Schwarzian criterion for univalence holds in $Ω$ if and only if the components of $\partialΩ$ are either points or quasicircles. We generalize this theorem to harmonic mappings.
