Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

Article

HOME ALL ARTICLES View

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.

Bj\"{o}rling formula for mean curvature one surfaces in hyperbolic three-space and in de Sitter three-space

Seong-Deog Yang

Korea University

Abstract

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