Complete bounded null curves immersed in {\mathbb {C}^3} and {\rm {SL}(2,\mathbb {C})} |
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Authors: | Francisco Martin Masaaki Umehara Kotaro Yamada |
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Institution: | 1. Departamento de Geometría y Topología, Universidad de Granada, 18071, Granada, Spain 2. Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka, 560-0043, Japan 3. Faculty of Mathematics, Kyushu University, Fukuoka, 812-8581, Japan
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Abstract: | We construct a simply connected complete bounded mean curvature one surface in the hyperbolic 3-space ${\mathcal {H}^3}$ . Such a surface in ${\mathcal {H}^3}$ can be lifted as a complete bounded null curve in ${\rm {SL}(2,\mathbb {C})}$ . Using a transformation between null curves in ${\mathbb {C}^3}$ and null curves in ${\rm {SL}(2,\mathbb {C})}$ , we are able to produce the first examples of complete bounded null curves in ${\mathbb {C}^3}$ . As an application, we can show the existence of a complete bounded minimal surface in ${\mathbb {R}^3}$ whose conjugate minimal surface is also bounded. Moreover, we can show the existence of a complete bounded immersed complex submanifold in ${\mathbb {C}^2}$ . |
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