Punctured Spheres

Complete minimal surfaces defined on punctured spheres with finite total curvature have meromorphic Weierstrass data and hence can be obtained by integrating rational functions. This makes them very accessible. On the other hand, there are only very few embedded examples.

k-Noids (all ends are catenoidal)

• Catenoid
• Jorge-Meeks k-Noids
• 3-Noids
• 4-Noids
• k-Noids with one vertical end
• k-Noids with two vertical ends

Planar and catenoidal ends

• Finite Riemann
• Finite Riemann with 4 ends
• Connor's Vase

Higher Order ends

• Enneper-Planar
• Double Enneper
• Wavy Catenoid
• Enepper with two Catenoids
• Longhurst