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)
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(2)
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where is the Weierstrass elliptic function with parameters
(3)
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(4)
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with the imaginary unit (and where turns out to be real and positive), and the constant given by
(5)
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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)
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(7)
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(8)
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where is a small positive constant and .
The second Chen-Gackstatter surface has topological genus and total curvature . Its Enneper-Weierstrass parameterization is
(9)
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(10)
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where , , and are positive numbers such that and, given the definitions
(11)
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(12)
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(13)
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(14)
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it holds that
(15)
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and
(16)
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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)
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(18)
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where
(19)
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and , are suitable real numbers. They can be chosen in such a way that the triple
(20)
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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.