Driving functions and traces of the Loewner equation |
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Authors: | HaiHua Wu XinHan Dong |
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Institution: | 1. College of Mathematics and Computer Science, Key Laboratory of High Performance Computing and Stochastic Information Processing, Hunan Normal University, Changsha, 410081, China
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Abstract: | We consider the chordal Loewner differential equation in the upper half-plane, the behavior of the driving function λ(t) and the generated hull K t when K t approaches λ(0) in a fixed direction or in a sector. In the case that the hull K t is generated by a simple curve γ(t) with γ(0) = 0, we prove some sharp relations of \({{\lambda (t)} \mathord{\left/ {\vphantom {{\lambda (t)} {\sqrt t }}} \right. \kern-0em} {\sqrt t }}\) and \({{\gamma (t)} \mathord{\left/ {\vphantom {{\gamma (t)} {\sqrt t }}} \right. \kern-0em} {\sqrt t }}\) as t → 0 which improve the previous work. |
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Keywords: | Loewner equation hull half-plane capacity driving function |
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