Some Pictures

The surfaces on this page belong to the family of Weingarten  surfaces in  either an  euclidean or hyperbolic 3-space. They have been studied in several papers of J. A. Gálvez, A. Martínez and F. Milán. We shall use the following notation:

  1. Flat surface: surface in the hyperbolic 3-space with vanishing Gaussian curvature.

  2. W-surface: surface in the hyperbolic 3-space whose mean curvature and Gaussian curvature  satisfy a non trivial linear relation.

  3. K-surface: surface of constant Gaussian curvature K.

 

Cylinder

Complete flat surface of revolution

Horosphere

Complete flat surface of revolution

Horosphere

Complete flat surface of revolution

non complete flat surface of revolution

non complete flat surface of revolution

non complete flat surface of revolution

non complete flat surface of revolution

Flat surface with an irregular end.

Flat surface with a multivaluate end

Non embedded complete W-surface with embedded asymptotic boundary

Complete W-surface with non regular asymptotic boundary

Complete totally umbilical W-surface

Complete W-surface with a non connected embedded asymptotic boundary

K-surface of revolution in the Euclidean 3-space, K>0.

K-surface of revolution in the Euclidean 3-space, K>0.

K-surface of revolution in the Euclidean 3-space, K<0.

K-surface of revolution in the Euclidean 3-space, K<0.

Pseudosphere

K-surface of revolution in the Euclidean 3-space, K<0.

Dini's surface

K-surface in the Euclidean 3-space, K<0.

K-surface of revolution in the hyperbolic 3-space, -1<K<0.

K-surface of revolution in the hyperbolic 3-space, K>0.