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]

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

References

^ Chen, Chi Cheng; Gackstatter, Fritz (1982), "Elliptische und hyperelliptische Funktionen und vollständige Minimalflächen vom Enneperschen Typ", Math. Ann., 259 (3): 359–369, doi:10.1007/bf01456948, S2CID 120602853
^ ^a ^b Thayer, Edward C. (1995), "Higher-genus Chen–Gackstatter surfaces and the Weierstrass representation for surfaces of infinite genus", Experiment. Math., 4 (1): 19–39, doi:10.1080/10586458.1995.10504305
^ Barile, Margherita. "Chen–Gackstatter Surfaces". MathWorld.
^ López, F. J. (1992), "The classification of complete minimal surfaces with total curvature greater than −12π", Trans. Amer. Math. Soc., 334: 49–73, doi:10.1090/s0002-9947-1992-1058433-9.