Enneper surface

Short description: Minimal surface

A portion of the Enneper surface

In differential geometry and algebraic geometry, the Enneper surface is a self-intersecting surface that can be described parametrically by: x=13u(1−13u2+v2),y=13v(1−13v2+u2),z=13(u2−v2). It was introduced by Alfred Enneper in 1864 in connection with minimal surface theory.[1][2][3][4]

The Weierstrass–Enneper parameterization is very simple, f(z)=1,g(z)=z, and the real parametric form can easily be calculated from it. The surface is conjugate to itself.

Implicitization methods of algebraic geometry can be used to find out that the points in the Enneper surface given above satisfy the degree-9 polynomial equation 64z9−128z7+64z5−702x2y2z3−18x2y2z+144(y2z6−x2z6)+162(y4z2−x4z2)+27(y6−x6)+9(x4z+y4z)+48(x2z3+y2z3)−432(x2z5+y2z5)+81(x4y2−x2y4)+240(y2z4−x2z4)−135(x4z3+y4z3)=0.

Dually, the tangent plane at the point with given parameters is a+bx+cy+dz=0, where a=−(u2−v2)(1+13u2+13v2),b=6u,c=6v,d=−3(1−u2−v2). Its coefficients satisfy the implicit degree-6 polynomial equation 162a2b2c2+6b2c2d2−4(b6+c6)+54(ab4d−ac4d)+81(a2b4+a2c4)+4(b4c2+b2c4)−3(b4d2+c4d2)+36(ab2d3−ac2d3)=0.

The Jacobian, Gaussian curvature and mean curvature are J=181(1+u2+v2)4,K=−491J,H=0. The total curvature is −4π. Osserman proved that a complete minimal surface in ℝ3 with total curvature −4π is either the catenoid or the Enneper surface.[5]

The curves obtained by setting u=0 or v=0 are geodesics of the Enneper surface, and are copies of the Tschirnhausen cubic.[6]

Another property is that all bicubical minimal Bézier surfaces are, up to an affine transformation, pieces of the surface.[7]

It can be generalized to higher order rotational symmetries by using the Weierstrass–Enneper parameterization f(z)=1,g(z)=zk for integer k>1.[3] It can also be generalized to higher dimensions; Enneper-like surfaces are known to exist in ℝn for n up to 7.[8]

See also [9] [10] for higher order algebraic Enneper surfaces.

## References

1. ↑ J.C.C. Nitsche, "Vorlesungen über Minimalflächen", Springer (1975)
2. ↑ Francisco J. López, Francisco Martín, Complete minimal surfaces in R3
3. ↑ 3.0 3.1 Ulrich Dierkes, Stefan Hildebrandt, Friedrich Sauvigny (2010). Minimal Surfaces. Berlin Heidelberg: Springer. ISBN 978-3-642-11697-1.
4. ↑ Weisstein, Eric W.. "Enneper's Minimal Surface". http://mathworld.wolfram.com/EnnepersMinimalSurface.html.
5. ↑ R. Osserman, A survey of Minimal Surfaces. Vol. 1, Cambridge Univ. Press, New York (1989).
6. ↑ Maksimović, Miroslav; Jovanović, Tanja; Ljajko, Eugen; Ivanović, Milica (2020). "Analysis of geodesics on different surfaces". Bulletin of Natural Sciences Research 10 (1): 51–56. doi:10.5937/univtho10-20589.
7. ↑ Cosín, C., Monterde, Bézier surfaces of minimal area. In Computational Science — ICCS 2002, eds. J., Sloot, Peter, Hoekstra, Alfons, Tan, C., Dongarra, Jack. Lecture Notes in Computer Science 2330, Springer Berlin / Heidelberg, 2002. pp. 72-81 ISBN 978-3-540-43593-8
8. ↑ Jaigyoung Choe, On the existence of higher dimensional Enneper's surface, Commentarii Mathematici Helvetici 1996, Volume 71, Issue 1, pp 556-569
9. ↑ E. Güler, Family of Enneper minimal surfaces. Mathematics. 2018; 6(12):281. https://doi.org/10.3390/math6120281
10. ↑ E. Güler, The algebraic surfaces of the Enneper family of maximal surfaces in three dimensional Minkowski space. Axioms. 2022; 11(1):4. https://doi.org/10.3390/axioms11010004

## External links

* Hazewinkel, Michiel, ed. (2001), "Enneper surface", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=p/e035710
* Mathematical Sciences Research Institute - The Generalized Enneper's Surfaces (Archived)
* Harvey Mudd College - Enneper's Surface (Archived)

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Minimal surfaces

* Associate family
* Bour's
* Catalan's
* Catenoid
* Chen–Gackstatter
* Costa's
* Enneper
* Gyroid
* Helicoid
* Henneberg
* k-noid
* Lidinoid
* Neovius
* Richmond
* Riemann's
* Saddle tower
* Scherk
* Schwarz
* Triply periodic

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