齿轮、预齿轮及相关区域
Gears, Pregears and Related Domains
AI总结：
本文从 Schwarzian 导数视角研究单位圆盘到单齿齿轮状区域的共形映射，引入“预齿轮”概念并精确确定了一致性区域。
摘要：
我们从 Schwarzian 导数的视角，研究从单位圆盘到单齿齿轮状平面区域的共形映射。齿轮状（或“类齿轮”）区域属于我们称之为“预齿轮”（齿轮在 Mobius 变换下的像）的更一般区域范畴，这有助于研究齿轮的共形映射，我们也对此进行了详细描述。由于此类区域由圆弧围成，已知 Riemann 映射的 Schwarzian 导数是具有特定形式的有理函数。这些映射的一个辅助参数自然地与齿轮（或预齿轮）的共形模相关，我们证明了若干将其与另一个主要剩余辅助参数相关联的定性结果。我们精确确定了对应的一致性区域（使该有理函数成为共形映射 Schwarzian 导数的参数取值范围）。
英文摘要：
We study conformal mappings from the unit disk to one-toothed gear-shaped planar domains from the point of view of the Schwarzian derivative. Gear-shaped (or "gearlike") domains fit into a more general category of domains we call "pregears" (images of gears under Mobius transformations), which aid in the study of the conformal mappings for gears and which we also describe in detail. Such domains being bounded by arcs of circles, the Schwarzian derivative of the Riemann mapping is known to be a rational function of a specific form. One accessory parameter of these mappings is naturally related to the conformal modulus of the gear (or pregear) and we prove several qualitative results relating it to the principal remaining accessory parameter. The corresponding region of univalence (parameters for which the rational function is the Schwarzian derivative of a conformal mapping) is determined precisely.
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