Complete flat surfaces with two isolated singularities in hyperbolic 3-space (Q961057)

The goal of this article is to study the family of complete flat surfaces with two isolated singularities in hyperbolic 3-space. Contents: importance of the subject, flat surfaces in \(H^3\) with isolated singularities, canonical examples (notational examples, examples with two isolated singularities) and characterizations. The authors classify complete embedded flat surfaces in \(H^3\) with either one or two isolated singularities and only one end. Theorem 1: ``The revolution examples are the unique complete flat embeddings in \(H^3\) with only one isolated singularity and one end.'' The result is complete by Theorem 2: ``Each complete flat embedding in \(H^3\) with only two isolated singularities and one end must be congruent to one of the canonical examples.'' For the proof the authors construct standard examples of flat surfaces in \(H^3\) which are graphs over a two-punctured horosphere. Theorem 3: ``Each canonical example \(\psi:\mathbb{A}_r\setminus\{z_0\}\to H^3\) is a complete flat embedding with two isolated singularities and one end''. Finally, a certain Appendix is presented. The method use is constructive.