“Chen–Gackstatter surface”的意思、由来-开放百科全书

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

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

1. ^{{citation|last1=Chen|first1=Chi Cheng|last2=Gackstatter|first2=Fritz|title=Elliptische und hyperelliptische Funktionen und vollständige Minimalflächen vom Enneperschen Typ|journal=Math. Ann.|volume=259|pages=359–369|year=1982|url=http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002321963&IDDOC=129137|doi=10.1007/bf01456948}}
2. ^¹{{citation|first=Edward C.|last=Thayer|title=Higher-genus Chen–Gackstatter surfaces and the Weierstrass representation for surfaces of infinite genus|journal=Experiment. Math.|volume=4|issue=1|year=1995|pages=19–39|url=http://projecteuclid.org/euclid.em/1062621140|doi=10.1080/10586458.1995.10504305}}
3. ^{{mathworld|author=Barile, Margherita|title=Chen–Gackstatter Surfaces|urlname=Chen-GackstatterSurfaces}}
4. ^{{citation|last=López|first=F. J.|title=The classification of complete minimal surfaces with total curvature greater than −12π|journal=Trans. Amer. Math. Soc.|volume=334|pages=49–73|year=1992|doi=10.1090/s0002-9947-1992-1058433-9}}.

References

External links