Constructing mean curvature 1 surfaces in $H^3$ with irregular ends

zbMATH Open1100.53051arXiv0805.3751MaRDI QIDQ5486616

Wayne Rossman, Masaaki Umehara, Kotaro Yamada

Publication date: 11 September 2006

Abstract: With the developments of the last decade on complete constant mean curvature 1 (CMC 1) surfaces in the hyperbolic 3-space

H^{3}

, many examples of such surfaces are now known. However, most of the known examples have regular ends. (An end is irregular, resp. regular, if the hyperbolic Gauss map of the surface has an essential singularity, resp. at most a pole, there.) There are some known surfaces with irregular ends, but they are all either reducible or of infinite total curvature. (The surface is reducible if and only if the monodromy of the secondary Gauss map can be simultaneously diagonalized.) Up to now there have been no known complete irreducible CMC 1 surfaces in

H^{3}

with finite total curvature and irregular ends. The purpose of this paper is to construct countably many 1-parameter families of genus zero CMC 1 surfaces with irregular ends and finite total curvature, which have either dihedral or Platonic symmetries. For all the examples we produce, we show that they have finite total curvature and irregular ends. For the examples with dihedral symmetry and the simplest example with tetrahedral symmetry, we show irreducibility. Moreover, we construct a genus one CMC 1 surface with four irregular ends, which is the first known example with positive genus whose ends are all irregular.

Full work available at URL: https://arxiv.org/abs/0805.3751

zbMATH Keywords

surface hyperbolic space constant mean curvature end

Mathematics Subject Classification ID

Minimal surfaces in differential geometry, surfaces with prescribed mean curvature (53A10) Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) (53C42)