Boy's Surface

Roman Gietz, Oliver Gross



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]


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

Roman Gietz   +

University: TU Berlin
E-Mail: gietz[at]

Oliver Gross   +

University: TU Berlin
E-Mail: ogross[at]