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, C. C.; Gackstatter, F. (1982), "Elliptische und hyperelliptische Funktionen und vollständige Minimalflächen vom Enneperschen Typ", Math. Ann. 259: 359–369, doi:10.1007/bf01456948
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.

External links

The Chen–Gackstatter Thayer Surfaces at the Scientific Graphics Project [1]
Chen–Gackstatter Surface in the Minimal Surface Archive [2]
Xah Lee's page on Chen–Gackstatter [3]

Mathematics Encyclopedia