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Chen-Gackstatter Surfaces


A class of complete orientable minimal surfaces of R^3 derived from Enneper's minimal surface. They are named for the mathematicians who found the first two examples in 1982.

The Chen-Gackstatter surfaces form a double-indexed collection M_(ij), where i>=0 and j>=1. M_(0,1) is Enneper's minimal surface, and M_(i1) is obtained from M_(0,1) by adding i handles so that it has topological genus equal to i. It has one Enneper end with winding order three, meaning that, like Enneper's minimal surface, it has a symmetric three-fold shape which tends to coincide with a triple plane far away from the center.

In general, M_(ij) has total curvature c=-4pi(i+1)j, topological genus ij, and one Enneper end of winding order 2j+1. This property distinguishes it from other surfaces such as the catenoid which have two ends of winding order 1.

The first Chen-Gackstatter surface M_(1,1) has topological genus p=1 and total curvature -8pi. Its Enneper-Weierstrass parameterization is given by

g(z)=(AP^'(z))/(P(z))
(1)
f(z)=2P(z),
(2)

where P(z) is the Weierstrass elliptic function with parameters

g_2=60sum^'_(m,n=-infty)^infty1/((m+ni)^4)
(3)
g_3=0,
(4)

with i the imaginary unit (and where g_2 turns out to be real and positive), and the constant A given by

 A=sqrt((3pi)/(2g_2)).
(5)

López (1992) has shown that M_(1,1) is the only genus one orientable complete minimal surface of total curvature -8pi.

In a neighborhood of the origin M_(11) can be approximated by the following parametric equations:

x=(4A^2)/(3r^3)cos(3theta)
(6)
y=-(4A^2)/(3r^3)sin(3theta)
(7)
z=(2A)/(r^2)cos(2theta),
(8)

where r is a small positive constant and 0<=theta<=2pi.

The second Chen-Gackstatter surface M_(2,1) has topological genus p=2 and total curvature -12pi. Its Enneper-Weierstrass parameterization is

g(z)=B(sqrt(z(z^2-a^2)(z^2-b^2)))/(z^2-a^2)
(9)
f(z)=(z^2-a^2)/(sqrt(z(z^2-a^2)(z^2-b^2))),
(10)

where a, b, and B are positive numbers such that a<b and, given the definitions

F_1=int_0^a(a^2-x^2)/(sqrt(x(a^2-x^2)(b^2-x^2)))dx
(11)
F_2=int_0^a(x(b^2-x^2))/(sqrt(x(a^2-x^2)(b^2-x^2)))dx
(12)
F_3=int_a^b(x^2-a^2)/(sqrt(x(x^2-a^2)(b^2-x^2)))dx
(13)
F_4=int_a^b(x(b^2-x^2))/(sqrt(x(x^2-a^2)(b^2-x^2)))dx,
(14)

it holds that

 F_1=B^2F_2,
(15)

and

 F_1F_4=F_2F_3.
(16)

The surfaces M_(1j) and M_(2j) were classified by Karcher (1989) and Thayer (1995), respectively. Sato (1996) completed the work for all M_(ij), and proved that the Enneper-Weierstrass parameterization of M_(ij) is given by

g=cw^i
(17)
f=1/g
(18)

where

 w^(i+1)=(zproduct_(1<=k<=j/2)(z^2-a_(2k)^2))/(product_(1<=l<=(j+1)/2)(z^2-a_(2l-1)^2)),
(19)

and c, a_1,...,a_j are suitable real numbers. They can be chosen in such a way that the triple

 ((1/g-g)dh,i(1/g+g)dh,2dh)
(20)

does not depend on the value of w.

Chen-Gackstatter surfaces

The pictures above (Hoffman et al. ) visualize the role of the double-index: M_(ij) has i holes along its axis of symmetry, which is surrounded by a curly rim with j+1 mountains and valleys.


This entry contributed by Margherita Barile

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References

Chen, C. C. and Gackstatter, F. "Elliptische und hyperelliptische Funktionen und vollständige Minimalflächen vom Enneperschen Typ." Math. Ann. 259, 359-369, 1982.Do Spirito-Santo, N. "Complete Minimal Surfaces in R^3 with Type Enneper End." Ann. Inst. Fourier 44, 525-557, 1994.GRAPE. "Chen-Gackstatter Surface." http://www-sfb256.iam.uni-bonn.de/grape/EXAMPLES/AMANDUS/gackstatter.html.Hoffman, J. T. et al. "The Chen-Gackstatter Thayer Surfaces." http://www.msri.org/publications/sgp/jim/geom/minimal/library/chengack/main.html.Karcher, H. "Construction of Minimal Surfaces." In Surveys in Geometry. University of Tokyo, pp. 1-96, 1989.López, F. J. "The Classification of Complete Minimal Surfaces with Total Curvature Greater than -12pi." Trans. Amer. Math. Soc. 334, 49-73, 1992.Sato, K. "Construction of Higher Genus Minimal Surfaces with One End and Finite Total Curvature." Tôhoku Math. J. 48, 229-246, 1996.Thayer, E. C. "Higher-Genus Chen-Gackstatter Surfaces and The Weierstrass Representation for Surfaces of Infinite Genus." Exper. Math. 4, 19-39, 1995.

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Chen-Gackstatter Surfaces

Cite this as:

Barile, Margherita. "Chen-Gackstatter Surfaces." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/Chen-GackstatterSurfaces.html

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