Communications in Analysis and Geometry

Volume 17 (2009)

Number 3

Spacelike mean curvature one surfaces in de Sitter $3$-space

Pages: 383 – 427



Shoichi Fujimori (Department of Mathematics, Fukuoka University of Education, Japan)

Wayne Rossman (Department of Mathematics, Faculty of Science, Kobe University, Japan)

Masaaki Umehara (Department of Mathematics, Graduate School of Science, Osaka University, Japan)

Kotaro Yamada (Department of Mathematics, Tokyo Institute of Technology)

Seong-Deog Yang (Department of Mathematics, Korea University, Seoul)


The first author studiedspacelike constant mean curvature one (CMC-$1$)surfaces in the de Sitter 3-space $S^3_1$when the surfaces have no singularities except within somecompact subsets and are of finite total curvature onthe complement of this compact subset.However, there are many CMC-$1${} surfaces whosesingular sets are not compact.In fact, such examples have already appeared in theconstruction of trinoids given by Lee and the last authorvia hypergeometric functions.

In this paper, we improve the Osserman-type inequalitygiven by the first author.Moreover, we shall develop a fundamental framework that allowsthe singular set to be non-compact, and thenwill use it to investigate the global behavior of CMC-$1${} surfaces.

Published 1 January 2009