A hyperbolic minimal surface is the image of an immersion into 3-dimensional hyperbolic space whose mean curvature vanishes. Totally geodesic hyperbolic planes are the simplest examples.
When the ambient sectional curvature is normalized to 
,
the Gauss-Codazzi equations imply
 |
where 
is the Gaussian curvature and 
are the principal
curvatures. The second equality uses 
, which follows from zero mean
curvature.
For an oriented conformal immersion, the Hopf differential is holomorphic. Away from its zeros, a conformal coordinate can be normalized so that the Gauss-Codazzi equations reduce to a cosh-Gordon equation. Hyperbolic minimal surfaces can also be constructed by the DPW method. Under the conformal compactification of 3-dimensional hyperbolic space, hyperbolic minimal surfaces satisfy the Willmore surface equation. Suitable complete minimal surfaces extend across the ideal boundary to Willmore surfaces in the 3-sphere (Bobenko et al. 2019).
-Noids
in Hyperbolic and Anti-de Sitter 3-Space." Proc. Roy. Soc. A 475,
20190173, 2019. 
via Willmore Geometry, Gauss Maps, Meromorphic Potentials, and Algorithmic Constructions."
Open Math. 24, 20250246, 2026.