A class of complete orientable minimal surfaces of 
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 
, where 
and 
. 
is Enneper's
minimal surface, and 
is obtained from 
by adding 
handles so that it has topological genus equal to 
. 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, 
has total curvature 
, topological genus 
, and one Enneper end of winding order 
. This property distinguishes it from other surfaces such
as the catenoid which have two ends of winding order
1.
The first Chen-Gackstatter surface 
has topological genus 
and total curvature 
. Its Enneper-Weierstrass
parameterization is given by
 |  |  |
(1)
|
 |  |  |
(2)
|
where 
is the Weierstrass elliptic function
with parameters
 |  |  |
(3)
|
 |  |  |
(4)
|
with 
the imaginary unit (and where 
turns out to be real and positive), and the constant 
given by
 |
(5)
|
López (1992) has shown that 
is the only genus one orientable complete minimal surface
of total curvature 
.
In a neighborhood of the origin 
can be approximated by the following parametric equations:
 |  |  |
(6)
|
 |  |  |
(7)
|
 |  |  |
(8)
|
where 
is a small positive constant and 
.
The second Chen-Gackstatter surface 
has topological genus 
and total curvature 
. Its Enneper-Weierstrass
parameterization is
 |  |  |
(9)
|
 |  |  |
(10)
|
where 
,
,
and 
are positive numbers such that 
and, given the definitions
 |  |  |
(11)
|
 |  |  |
(12)
|
 |  |  |
(13)
|
 |  |  |
(14)
|
it holds that
 |
(15)
|
and
 |
(16)
|
The surfaces 
and 
were classified by Karcher (1989) and Thayer (1995), respectively. Sato (1996) completed
the work for all 
, and proved that the Enneper-Weierstrass parameterization
of 
is given by
 |  |  |
(17)
|
 |  |  |
(18)
|
where
 |
(19)
|
and 
,
are suitable real numbers. They can be chosen in such a way that the triple
 |
(20)
|
does not depend on the value of 
.
The pictures above (Hoffman et al. ) visualize the role of the double-index: 
has 
holes along its axis of symmetry, which is surrounded by a curly rim with 
mountains and valleys.
with Type Enneper End." Ann. Inst. Fourier 44,
525-557, 1994.GRAPE. "Chen-Gackstatter Surface." 
." 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.