论文标题
通过Loewner微分方程在双连接域上的挤压功能
The squeezing function on doubly-connected domains via the Loewner differential equation
论文作者
论文摘要
对于任何有限的域，$ \ mathbb {c}^{n} $中的$ω$，邓，u和zhang引入了挤压函数$s_Ω（z）$，这是一个有界域的生物形态不变的。我们表明，对于$ n = 1 $，在\ mathbb {c}中挤压函数$ a_r = \ lbrace z \：r <| z | <1 \ rbrace $由$ s_ {a_r}（z）= \ max \ left \ lbrace | z |给出，\ frac {r} {| z |} \ right \ rbrace $ for ash $ 0 <r <1 $。这否认了邓，关和张提出的挤压功能的猜想公式，并建立了（直至生物形态学）在$ \ mathbb {c} $以外的所有双连接域的挤压函数以外的所有双连接域。它为宽类平面域的挤压功能提供了第一个非平凡公式，并回答了一个问题。我们用于证明此结果的主要工具是Schottky-Klein Prime功能（遵循Crowdy的工作）和由于Komatu而在Annuli上的Loewner微分方程版本。我们还表明，这些结果可用于在$ \ mathbb {c}^{n} $中的某些产品域的挤压功能上获得下限。
For any bounded domains $Ω$ in $\mathbb{C}^{n}$, Deng, Guan and Zhang introduced the squeezing function $S_Ω(z)$ which is a biholomorphic invariant of bounded domains. We show that for $n=1$, the squeezing function on an annulus $A_r = \lbrace z \in \mathbb{C} : r <|z| <1 \rbrace$ is given by $S_{A_r}(z)= \max \left\lbrace |z| ,\frac{r}{|z|} \right\rbrace$ for all $0<r<1$. This disproves the conjectured formula for the squeezing function proposed by Deng, Guan and Zhang and establishes (up to biholomorphisms) the squeezing function for all doubly-connected domains in $\mathbb{C}$ other than the punctured plane. It provides the first non-trivial formula for the squeezing function for a wide class of plane domains and answers a question of Wold. Our main tools used to prove this result are the Schottky-Klein prime function (following the work of Crowdy) and a version of the Loewner differential equation on annuli due to Komatu. We also show that these results can be used to obtain lower bounds on the squeezing function for certain product domains in $\mathbb{C}^{n}$.
