Enneper surface

In differential geometry and algebraic geometry, the Enneper surface is a self-intersecting surface that can be described parametrically by: ${\begin{aligned}x&={\tfrac {1}{3}}u\left(1-{\tfrac {1}{3}}u^{2}+v^{2}\right),\\y&={\tfrac {1}{3}}v\left(1-{\tfrac {1}{3}}v^{2}+u^{2}\right),\\z&={\tfrac {1}{3}}\left(u^{2}-v^{2}\right).\end{aligned}}$ 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 ${\begin{aligned}&64z^{9}-128z^{7}+64z^{5}-702x^{2}y^{2}z^{3}-18x^{2}y^{2}z+144(y^{2}z^{6}-x^{2}z^{6})\\&{}+162(y^{4}z^{2}-x^{4}z^{2})+27(y^{6}-x^{6})+9(x^{4}z+y^{4}z)+48(x^{2}z^{3}+y^{2}z^{3})\\&{}-432(x^{2}z^{5}+y^{2}z^{5})+81(x^{4}y^{2}-x^{2}y^{4})+240(y^{2}z^{4}-x^{2}z^{4})-135(x^{4}z^{3}+y^{4}z^{3})=0.\end{aligned}}$

Dually, the tangent plane at the point with given parameters is $a+bx+cy+dz=0,\$ where ${\begin{aligned}a&=-\left(u^{2}-v^{2}\right)\left(1+{\tfrac {1}{3}}u^{2}+{\tfrac {1}{3}}v^{2}\right),\\b&=6u,\\c&=6v,\\d&=-3\left(1-u^{2}-v^{2}\right).\end{aligned}}$ Its coefficients satisfy the implicit degree-6 polynomial equation ${\begin{aligned}&162a^{2}b^{2}c^{2}+6b^{2}c^{2}d^{2}-4(b^{6}+c^{6})+54(ab^{4}d-ac^{4}d)+81(a^{2}b^{4}+a^{2}c^{4})\\&{}+4(b^{4}c^{2}+b^{2}c^{4})-3(b^{4}d^{2}+c^{4}d^{2})+36(ab^{2}d^{3}-ac^{2}d^{3})=0.\end{aligned}}$

The Jacobian, Gaussian curvature and mean curvature are ${\begin{aligned}J&={\frac {1}{81}}(1+u^{2}+v^{2})^{4},\\K&=-{\frac {4}{9}}{\frac {1}{J}},\\H&=0.\end{aligned}}$ The total curvature is $-4\pi$ . Osserman proved that a complete minimal surface in $\mathbb {R} ^{3}$ with total curvature $-4\pi$ is either the catenoid or the Enneper surface.^[5]

Another property is that all bicubical minimal Bézier surfaces are, up to an affine transformation, pieces of the surface.^[6]

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

See also ^[8] ^[9] for higher order algebraic Enneper surfaces.

References

^ J.C.C. Nitsche, "Vorlesungen über Minimalflächen", Springer (1975)
^ Francisco J. López, Francisco Martín, Complete minimal surfaces in R3
^ ^a ^b Ulrich Dierkes, Stefan Hildebrandt, Friedrich Sauvigny (2010). Minimal Surfaces. Berlin Heidelberg: Springer. ISBN 978-3-642-11697-1.
^ Weisstein, Eric W. "Enneper's Minimal Surface". MathWorld.
^ R. Osserman, A survey of Minimal Surfaces. Vol. 1, Cambridge Univ. Press, New York (1989).
^ 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
^ Jaigyoung Choe, On the existence of higher dimensional Enneper's surface, Commentarii Mathematici Helvetici 1996, Volume 71, Issue 1, pp 556-569
^ E. Güler, Family of Enneper minimal surfaces. Mathematics. 2018; 6(12):281. https://doi.org/10.3390/math6120281
^ 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

[1] J.C.C. Nitsche, "Vorlesungen über Minimalflächen", Springer (1975)

[2] Francisco J. López, Francisco Martín, Complete minimal surfaces in R3

[dierkes-3] 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". MathWorld.

[5] R. Osserman, A survey of Minimal Surfaces. Vol. 1, Cambridge Univ. Press, New York (1989).

[6] 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

[7] Jaigyoung Choe, On the existence of higher dimensional Enneper's surface, Commentarii Mathematici Helvetici 1996, Volume 71, Issue 1, pp 556-569

[8] E. Güler, Family of Enneper minimal surfaces. Mathematics. 2018; 6(12):281. https://doi.org/10.3390/math6120281

[9] 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

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