Joint work with Nick Schmitt and Jonas Ziefle. Link to preprint.

These are minimal surfaces in the Euclidean space \(\mathbb{E}^3\) and CMC 1 surfaces in the hyperbolic space \(\mathbb{H}^3\). They are made using the Loop Weierstrass Representation.

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Euclidean space

No planar end

No planar end is the plain catenoid.

1 planar end

This case is made from the double cover of a catenoid.

2 planar ends

3 planar ends

Hyperbolic space

The surfaces are displayed in the Poincaré ball model of \(\mathbb{H}^3\).

No planar end

No planar end is the plain catenoid. But in the hyperbolic space, they come in a one-parameter family: (todo)

  • Embedded, large neck
  • Embedded, thin neck
  • Touching horospheres
  • Non-embedded, thin neck
  • Non-embedded, large neck

1 planar end

This case is tricky as it is made from the double cover of a catenoid.

2 planar ends (todo)

  • Vertical
  • Tilted

3 planar ends (todo)

  • Vertical
  • Tilted

Extras