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J. Korean Math. Soc. 2018; 55(5): 1235-1255

Online first article August 2, 2018      Printed September 1, 2018

https://doi.org/10.4134/JKMS.j170671

Copyright © The Korean Mathematical Society.

Helicoidal Killing fields, helicoids and ruled minimal surfaces in homogeneous three-manifolds

Young Wook Kim, Sung-Eun Koh, Hyung Yong Lee, Heayong Shin, Seong-Deog Yang

Korea University, Konkuk University, Korea University, Korea Institute for Advanced Study, Korea University

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We provide definitions for the helicoidal Killing field and the helicoid in arbitrary three-manifolds, and investigate helicoids and ruled minimal surfaces in homogeneous three-manifolds, mainly in $\rm{SL_2}\mathbb R$ and Sol(3). In so doing we finish our classification of ruled minimal surfaces in homogeneous three-manifolds with the isometry group of dimension 4.

Keywords: ruled minimal surfaces, special linear group, Killing fields, helicoids

MSC numbers: 53A10, 53A35, 53C30

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