# Boy's Surface

Roman Gietz, Oliver Gross

### Description

In 1901, under the supervision of David Hilbert, Werner Boy wrote his dissertation „Über die Curvatura integra und die Topologie geschlossener Flächen“[1]. It is noticeable that in this thesis the global formulation of the well-known Gauss-Bonnet theorem appeared for the first time.

In particular his work contained a discussion of the theorem for non-orientable surfaces. This lead to a construction of an immersion of the real projective plane $\mathbb{R}P^2$ into $\mathbb{R}^3$ which was, until then, thought to be unlikely possible[2].

Rob Kusner and Robert Bryant found a particularly appealing parameterisation of the so called „Boy’s surface“ [3] which minimises the „Willmore functional“ $$W(f)=\frac{1}{4}\int_M (\kappa_1 + \kappa_2)^2\ dA\ .$$
Willmore surfaces are perceived to be exceptionally attractive as they do not possess any unnecessary dents.

Boy’s surface is related to Kusner surfaces as you can obtain the former from a Kusner surface of degree 3 by applying a suitable sphere inversion.[3]

### References

• Werner Boy.
Über die Curvatura integra und die Topologie geschlossener Flächen.
Mathematische Annalen, 57(2):151–184, June 1903.
doi:10.1007/BF01444342.
• Hermann Karcher and Ulrich Pinkall.
Die Boysehe Fläche in Oberwolfach.
Mitteilungen der DMV, Volume 5 (1), March 1997.
dgd:549.
• Rob Kusner.
Conformal Geometry and Complete Minimal Surfaces.
Bull. Amer. Math. Soc, 1987.
dgd:497.

#### Roman Gietz   +

University: TU Berlin
E-Mail: gietz[at]math.tu-berlin.de

#### Oliver Gross   +

University: TU Berlin
E-Mail: ogross[at]math.tu-berlin.de