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

Question

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.}

Answer 1

$\newcommand{\Reals}{\mathbf{R}}$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.

Answer 2

Individual saddle points, lines or rings can occur only locally, but it cannot occur for entire surfaces due to scalar curvature requirement:

$$ H=0, \kappa_{2} = -\kappa_{1} =1/a ; \, K= \kappa_{1} \kappa_{2} = -1/a^2 $$

To find where they occur let $$ \kappa_{2} = -\kappa_{1}= \tan \psi/a ; \, \kappa_n = 0 \, @ \psi= \pi/4 $$

The necessary condition is therefore that asymptotic lines cut orthogonally.

For example here, at $ \phi= 3 \pi/2 $ a slope line ring for Beltrami pseudosphere, \phi=0 for hyper_pseudospheres.

We may say that is locally valid :

1. Every minimal surface when sufficiently extended can contain a line of constant $K=-1, $ and,

2. Every K<0 surface has a line of H =0. $