In differential geometry, the ChenâGackstatter surface family (or the ChenâGackstatterâThayer 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 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 . [3] It has been shown that is the only genus one orientable complete minimal surface of total curvature . [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): 359â369, doi: 10.1007/bf01456948, S2CID 120602853
- ^ a b Thayer, Edward C. (1995), "Higher-genus ChenâGackstatter surfaces and the Weierstrass representation for surfaces of infinite genus", Experiment. Math., 4 (1): 19â39, doi: 10.1080/10586458.1995.10504305
- ^ Barile, Margherita. "ChenâGackstatter Surfaces". MathWorld.
- ^ LĂłpez, F. J. (1992), "The classification of complete minimal surfaces with total curvature greater than −12Ď", Trans. Amer. Math. Soc., 334: 49â73, doi: 10.1090/s0002-9947-1992-1058433-9.
External links
In differential geometry, the ChenâGackstatter surface family (or the ChenâGackstatterâThayer 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 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 . [3] It has been shown that is the only genus one orientable complete minimal surface of total curvature . [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): 359â369, doi: 10.1007/bf01456948, S2CID 120602853
- ^ a b Thayer, Edward C. (1995), "Higher-genus ChenâGackstatter surfaces and the Weierstrass representation for surfaces of infinite genus", Experiment. Math., 4 (1): 19â39, doi: 10.1080/10586458.1995.10504305
- ^ Barile, Margherita. "ChenâGackstatter Surfaces". MathWorld.
- ^ LĂłpez, F. J. (1992), "The classification of complete minimal surfaces with total curvature greater than −12Ď", Trans. Amer. Math. Soc., 334: 49â73, doi: 10.1090/s0002-9947-1992-1058433-9.