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 [math]\displaystyle{ M_{ij} }[/math] 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 [math]\displaystyle{ -4\pi(i+1)j }[/math].[3] It has been shown that [math]\displaystyle{ M_{11} }[/math] is the only genus one orientable complete minimal surface of total curvature [math]\displaystyle{ -8\pi }[/math].[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, http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002321963&IDDOC=129137
- ↑ 2.0 2.1 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, http://projecteuclid.org/euclid.em/1062621140
- ↑ Barile, Margherita. "Chen–Gackstatter Surfaces". http://mathworld.wolfram.com/Chen-GackstatterSurfaces.html.
- ↑ 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]
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