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With the developments of the last decade on complete constant mean curvature1 (CMC 1) surfaces in the hyperbolic 3-space $H^3$, many examples of suchsurfaces 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 surfacehas an essential singularity, resp. at most a pole, there.) There are someknown surfaces with irregular ends, but they are all either reducible or ofinfinite total curvature. (The surface is reducible if and only if themonodromy of the secondary Gauss map can be simultaneously diagonalized.) Up tonow there have been no known complete irreducible CMC 1 surfaces in $H^3$ withfinite total curvature and irregular ends. The purpose of this paper is toconstruct countably many 1-parameter families of genus zero CMC 1 surfaces withirregular ends and finite total curvature, which have either dihedral orPlatonic symmetries. For all the examples we produce, we show that they havefinite total curvature and irregular ends. For the examples with dihedralsymmetry and the simplest example with tetrahedral symmetry, we showirreducibility. Moreover, we construct a genus one CMC 1 surface with fourirregular ends, which is the first known example with positive genus whose endsare all irregular.

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