Bull. Korean Math. Soc. 2017; 54(1): 159-175
Online first article January 17, 2017 Printed January 31, 2017
https://doi.org/10.4134/BKMS.b150984
Copyright © The Korean Mathematical Society.
Seong-Deog Yang
Korea University
We solve the Bj\"{o}rling problem for constant mean curvature one surfaces in hyperbolic three-space and in de Sitter three-space. That is, we show that for any regular, analytic (and spacelike in the case of de Sitter three-space) curve $\gamma$ and an analytic (timelike in the case of de Sitter three-space) unit vector field $N$ along and orthogonal to $\gamma$, there exists a unique (spacelike in the case of de Sitter three-space) surface of constant mean curvature $1$ which contains $\gamma$ and the unit normal of which on $\gamma$ is $N$. Some of the consequences are the planar reflection principles, and a classification of rotationally invariant CMC $1$ surfaces.
Keywords: Bj\"{o}rling formula, constant mean curvature surfaces, de Sitter space
MSC numbers: 53B30
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