Q:

Do minimal hyperbolic surfaces exist? What do they look like?

I understand that it is impossible to embed* the entire hyperbolic plane in $\mathbb{R}^3$. But, can one create a embedding of part of the hyperbolic plane such that the resulting surface is also minimal? 
Basically, do there exist surfaces which have $\kappa_1 + \kappa_2 = 0$, but also $\kappa_1 \kappa_2 = const < 0$? What do they look like?
*"Embed" may not be the correct term here, but I hope the idea is clear.

A:

$\newcommand{\Reals}{\mathbf{R}}$There is no minimal surface of constant negative Gaussian curvature in $\Reals^{3}$, even locally.
Up to scaling, the principal curvatures would satisfy
$$
\kappa_{2} = -\kappa_{1},\qquad
-1 = \kappa_{1} \kappa_{2} = -\kappa_{1}^{2},
$$
so the principal curvatures would be constant: $\kappa_{1} = 1 = -\kappa_{2}$ without loss of generality.
If a surface in $\Reals^{3}$ has constant principal curvatures, the Codazzi equations give $\kappa_{1} - \kappa_{2} = 0$ or $\kappa_{1}\kappa_{2} = 0$. (See, for example, O'Neill, Elementary Differential Geometry, Second revised edition, Theorem 2.6, page 272.) This excludes a minimal surface of constant negative Gaussian curvature.
In case it's of interest, a connected surface in $\Reals^{3}$ having constant principal curvatures is part of a plane, cylinder, or sphere. See, for example, O'Neill, Elementary Differential Geometry, Second revised edition, Exercise 5 on page 280.

