Embedded Harmonic Surfaces

The surfaces in this section are not minimal, but are given by harmonic, non-conformal maps. They are all complete, properly embedded and have finite total Gauss curvature. This is a much larger class of surfaces than that of minimal surfaces. On one hand, the period conditions that need to be solved to get a closed minimal surface, can usually be trivially satisfied. On the other hand, there are many more possibilities for properly embedded ends of finite total curvature. However, the restriction to finite total curvature and properness imposes restrictions on the surfaces that are akin to Osserman's theorem for minimal surfaces.

Many classical obstructions for minimal surfaces disappear: For instance, there are embedded examples of any genus with just one end. Some limitations remain: Harmonic surfaces satisfy the maximum principle, and have non-positive Gauss curvature.

All surfaces here are the result of joint efforts of Peter Connor, Kevin Li, Adam Weyhaupt and myself.

Spheres

• Symmetric k-(0,0,1)
• Non-proper ends of type (0,0,n)
• Two ends of type (0,1,2)
• Two ends of type (0,1,n) and (0,1,2)
• Two ends of type (0,2,3)
• Six ends of type (0,2,3)
• Symmetric k-(0,1,2)s
• Two ends of type (0,0,1) and (2,3,4)
• Two ends of type (1,2,4)
• Two ends of type (2,2,n)
• Two ends of type (1,2,2) and one end of type (0,2,2)
• Alternating ends of type (0,0,1) and (0,1,2)
• One end of type (2,3,n)
• One end of type (2,3,2n)
• One end of type (2,4,2n)
• One end of type (2,5,8)
• One end of type (3,4,6)
• Two ends of type (1,2,n) and (0,0,1)
• Two ends of type (1,2,m) and (1,2,n)

Tori

• One end of type (2,2,3)
• Two ends of type (1,2,2) (catenoidal)
• Two ends of type (0,2,3)
• Two ends of type (2,2,3)

Higher Genus

• Genus two with two ends of type (1,2,2) (catenoidal)
• Genus two with one end of type (2,2,4)
• Genus two with three ends of type (2,2,3)
• Genus two and three with one end of type (2,2,3)
• Genus g with one end of type (2,2,g+2)