Scherk surface

2020 Mathematics Subject Classification: Primary: 53A10 Secondary: 49Q05 [MSN][ZBL]

A minimal surface discovered by H. Scherk in 1834. It is defined by the equation \[ z = \ln \frac{\cos y}{\cos x} = \ln \cos y - \ln \cos x \] and it is the only minimal surface that can be represented as a translation surface of the form $z = f(x) + g (y)$. Scherk's surface and its modifications are used for the construction of auxiliary functions that allow one to find examples of the unsolvability of the Dirichlet problem for the Euler–Lagrange equation for minimal surfaces over non-convex domains.

Scherk's surface possesses a number of interesting properties:

• it is a complete surface of infinite genus containing a countable number of straight lines;
• its universal covering surface presents an example of a complete minimal surface of conformally-hyperbolic type;
• its Gauss map omits exactly four points.

The last property of Scherk's surface becomes evident from its Weierstrass representation. More precisely if we consider the meromorphic function \[ w \mapsto f(w) = \frac{2}{1-w^4} \] and the holomorphic function $f (w) =w$, Scherk's surface can be parameterized as \[ \begin{array}{lll} x (r, \phi) &= \mathcal{Re}\, \int_0^\omega f (1-g^2)\, dw &= 2 \mathcal{Re}\, (ln (1+ \omega) - \ln (1- \omega))\\ y (r, \phi) &= \mathcal{Re}\, \int_0^\omega i f (1+g^2)\, dw & = \mathcal{Re}\, ( \tan^{-1} \omega)\\ z (r, \phi) &= \mathcal{Re}\, \int_0^\omega 2 f g\, dw & = 2 \mathcal{Re}\, [i (\ln (1+ \omega^2) - \ln (1-\omega^2))] \, , \end{array} \] where $\omega = r e^{i\phi}$. The four points missing in the image of the Gauss map corresponds then to the four poles $\{\pm 1, \pm i\}$ of $f$.

By analogy with this representation one may construct the generalized Scherk surfaces using \[ f (w) = [(w-w_1) (w-w_2) (w-w_3) (w-w_4)]^{-1} \] with arbitrary distinct points $w_1, w_2, w_3, w_4$ in the complex plane. The Gauss map of this minimal surface omits precisely $4$ arbitrarily assigned points on $\mathbb S^2$.

The existence of such minimal surfaces motivated the conjecture that there is no complete minimal surface in $\mathbb R^3$ whose (oriented) normal more than four directions. After several improvements of a classical result of Osserman, the conjecture was finally proved in [Fu]; see also [Xa].

Scherk's surface belongs to the family of so-called periodic minimal surfaces; a picture of it, and other interesting properties, can be found in [TF].

References

[DHKW]	U. Dierkes, S. Hildebrandt, A. Küster, O. Wohlrab, "Minimal surfaces", vol. I, p. 108. Springer 1992.
[DoC]	M.P. Do Carmo, "Differential geometry of curves and surfaces" , Prentice-Hall (1975) Zbl 0733.53001 Zbl 0606.53002 Zbl 0326.53001
[Fu]	H. Fujimoto, "On the number of exceptional values of the Gauss map of minimal surface." J. Math. Soc. Japan. 40 (1988) pp. 235-247
[Gr]	Gray, A. "Minimal Surfaces via the Weierstrass Representation." Ch. 32 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 735-760, 1997.
[Ni]	J.C.C. Nitsche, "Vorlesungen über Minimalflächen" , Springer (1975) MR0448224 Zbl 0319.53003
[Os]	R. Osserman., "A survey of minimal surfaces" , Dover (1986)
[Sp]	M. Spivak, "A comprehensive introduction to differential geometry" , 1979 , Publish or Perish MR0532834 MR0532833 MR0532832 MR0532831 MR0532830 MR0394453 MR0394452 MR0372756 MR1537051 MR0271845 MR0267467 Zbl 1213.53001 Zbl 0439.53005 Zbl 0439.53004 Zbl 0439.53003 Zbl 0439.53002 Zbl 0439.53001 Zbl 0306.53003 Zbl 0306.53002 Zbl 0306.53001 Zbl 0202.52201 Zbl 0202.52001
[St]	D.J. Struik, "Differential geometry" , Addison-Wesley (1957) MR0939369 MR1528072 MR0036551 MR1562092 Zbl 0697.53002 Zbl 0105.14707 Zbl 0041.48603 Zbl 0007.38806 Zbl 0001.16402 Zbl 59.0010.03 Zbl 57.0860.04
[TF]	A.A. Tuzhilin, A.T. Fomenko, "Elements of the geometry and topology of minimal surfaces" , Moscow (1991) (In Russian) MR1162118 MR1134130 Zbl 0745.53001 Zbl 0781.53002
[Xa]	F. Xavier, "The Gauss map of a complete non-at minimal surface cannot omit 7 points of the sphere." Ann. of Math. 113 (1981) pp. 211-214.