I construct a Weierstraß formula for surfaces of mean curvature one in hyperbolic space and use it to investigate the complete surfaces of mean curvature one and finite total curvature.
Although it was unknown to me at the time that I wrote this article, Bianchi had long ago pointed out that the local surfaces of mean curvature one in hyperbolic space admit a Weierstraß representation. (See Bianchi's Lezioni... Volume 2, Part 2, pp. 607-613.) While Bianchi's representation is not quite the same as the one I derive, the two are essentially equivalent for local purposes or in the simply connected case. I am grateful to Pedro Roitman for making me aware of Bianchi's work on this problem and supplying me with the above reference. In particular, it is clear that these CMC-1 surfaces in hyperbolic space should certainly not be called "Bryant surfaces".
Using my version of the Weierstraß formula, I show that, though each such surface is locally isometric to a minimal surface in Euclidean 3-space, there are striking differences. For example, even though such a surface is conformally a compact surface punctured at a finite number of points, the total area is not necessarily a rational multiple of Pi and the natural Gauss map need not complete holomorphically across the punctures. I compute some examples, investigate the simply connected case, and derive necessary and sufficient conditions for such a surface with prescribed seond fundamental form near a puncture to be realizable as a punctured disk in hyperbolic space.
Since this paper appeared, Umehara and others have computed more examples and further explored the geometry of these surfaces. The reader might try looking at works of
I do not have any reprints of this article (Asterisque never sent me any) and the original is a typescript, with no electronic version available.