## Journal of Differential Geometry

- J. Differential Geom.
- Volume 108, Number 3 (2018), 411-457.

### On the Björling problem for Willmore surfaces

#### Abstract

We solve the analogue of Björling’s problem for Willmore surfaces via a harmonic map representation. For the umbilic-free case the problem and solution are as follows: given a real analytic curve $y_0$ in $\mathbb{S}^3$, together with the prescription of the values of the surface normal and the dual Willmore surface along the curve, lifted to the light cone in Minkowski $5$-space $\mathbb{R}^5_1$, we prove, using isotropic harmonic maps, that there exists a unique pair of dual Willmore surfaces $y$ and $\hat{y}$ satisfying the given values along the curve. We give explicit formulae for the generalized Weierstrass data for the surface pair. For the three dimensional target, we use the solution to explicitly describe the Weierstrass data, in terms of geometric quantities, for all equivariant Willmore surfaces. For the case that the surface has umbilic points, we apply the more general half-isotropic harmonic maps introduced by Hélein to derive a solution: in this case the map $\hat{y}$ is not necessarily the dual surface, and the additional data of a derivative of $\hat{y}$ must be prescribed. This solution is generalized to higher codimensions.

#### Article information

**Source**

J. Differential Geom., Volume 108, Number 3 (2018), 411-457.

**Dates**

Received: 13 September 2014

First available in Project Euclid: 2 March 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.jdg/1519959622

**Digital Object Identifier**

doi:10.4310/jdg/1519959622

**Mathematical Reviews number (MathSciNet)**

MR3770847

**Zentralblatt MATH identifier**

06846982

#### Citation

Brander, David; Wang, Peng. On the Björling problem for Willmore surfaces. J. Differential Geom. 108 (2018), no. 3, 411--457. doi:10.4310/jdg/1519959622. https://projecteuclid.org/euclid.jdg/1519959622