Chen–Gackstatter surface

In differential geometry, the Chen–Gackstatter surface family (or the Chen–Gackstatter–Thayer surface family) is a family of minimal surfaces that generalize the Enneper surface by adding handles, giving it nonzero topological genus.^[1][2]

The first nine Chen–Gackstatter surfaces.

They are not embedded, and have Enneper-like ends. The members ${\displaystyle M_{ij}}$ of the family are indexed by the number of extra handles i and the winding number of the Enneper end; the total genus is ij and the total Gaussian curvature is ${\displaystyle -4\pi (i+1)j}$.^[3] It has been shown that ${\displaystyle M_{11}}$ is the only genus one orientable complete minimal surface of total curvature ${\displaystyle -8\pi }$.^[4]

It has been conjectured that continuing to add handles to the surfaces will in the limit converge to the Scherk's second surface (for j = 1) or the saddle tower family for j > 1.^[2]