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

TL;DR

This paper constructs new complete constant mean curvature 1 surfaces in hyperbolic 3-space with irregular ends, including the first positive genus example, expanding the known family of such surfaces with finite total curvature.

Contribution

It introduces the first known irreducible CMC 1 surfaces in $H^3$ with irregular ends and finite total curvature, including genus zero and genus one examples with specific symmetries.

Findings

Constructed countably many genus zero CMC 1 surfaces with irregular ends and symmetries.

Proved some examples are irreducible and have finite total curvature.

Presented the first genus one CMC 1 surface with irregular ends.

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.