Math @ Duke

Publications [#318278] of Robert Bryant
search www.ams.org.Papers Published
 Bryant, R, Surfaces of mean curvature one in hyperbolic space,
in Théorie des variétés minimales et applications (Palaiseau, 1983–1984), Astérisque, vol. 154155
(1988),
pp. 321347, Société Mathématique de France [MR955072]
(last updated on 2019/05/23)
Author's Comments: 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. 607613.) 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 CMC1 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
3space, 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
and the references contained in these papers.
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.


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