[Paper Review] On the Björling problem for Willmore surfaces

David Brander, Peng Wang|arXiv (Cornell University)|Sep 13, 2014

Geometric Analysis and Curvature Flows21 references3 citations

TL;DR

This paper solves the Björling problem for Willmore surfaces in $\mathbb{S}^3$ and higher codimensions using isotropic harmonic maps into $SO(1,4)/(SO(1,1)\times SO(3))$, extending the DPW method to construct unique dual Willmore surface pairs from analytic data on a curve. The key contribution is a generalized Weierstrass representation via holomorphic potentials, with explicit formulae for the surface pair and a solution for umbilic points using half-isotropic maps.

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 $S^3$, 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.

Motivation & Objective

• To extend the classical Björling problem—originally for minimal surfaces—to Willmore surfaces in conformal geometry.
• To establish a unique solution for Willmore surfaces given analytic initial data: a curve in $\mathbb{S}^3$, its normal, and the dual surface data along the curve.
• To develop a harmonic map-based construction using isotropic harmonic maps into $SO(1,4)/(SO(1,1)\times SO(3))$ for the DPW method.
• To generalize the solution to surfaces with umbilic points using half-isotropic harmonic maps, requiring derivative data of the dual map.
• To provide explicit Weierstrass data in terms of geometric quantities for equivariant Willmore surfaces in $\mathbb{R}^3$.

Proposed method

• Lift the surface and its dual to the light cone in $\mathbb{R}^5_1$, using the conformal Gauss map representation.
• Represent the surface pair via the isotropic harmonic map $Y \wedge \hat{Y}$, where $Y$ and $\hat{Y}$ are dual Willmore surfaces.
• Apply the DPW method to isotropic harmonic maps, using holomorphic frames and potentials in the loop group $\Lambda G$.
• Construct the extended frame via the Iwasawa decomposition of the holomorphic potential $\eta$.
• Verify that the DPW construction restricts to isotropic harmonic maps by checking the structure of the Maurer-Cartan form.
• For umbilic points, use Hélein’s half-isotropic harmonic maps, requiring prescription of $\hat{Y}$ and its derivative to ensure uniqueness.

Experimental results

Research questions

• RQ1 Can the Björling problem for Willmore surfaces be solved uniquely using conformal and harmonic map methods?
• RQ2 What additional data is required to ensure uniqueness in the presence of umbilic points?
• RQ3 How can the DPW method be adapted to isotropic harmonic maps for Willmore surface construction?
• RQ4 What is the explicit Weierstrass data for equivariant Willmore surfaces in $\mathbb{R}^3$?
• RQ5 How does the solution generalize from $\mathbb{S}^3$ to higher codimensions?

Key findings

• A unique pair of dual Willmore surfaces $y$ and $\hat{y}$ exists for any real analytic curve in $\mathbb{S}^3$, given the surface, its normal, and the dual surface data along the curve.
• The solution is constructed via the DPW method applied to isotropic harmonic maps, yielding explicit holomorphic potentials as Weierstrass data.
• For equivariant Willmore surfaces in $\mathbb{R}^3$, the Weierstrass data is expressed explicitly in terms of geometric quantities such as curvature and torsion of the initial curve.
• In the umbilic case, the dual map $\hat{y}$ is not necessarily the standard dual, and its derivative must be prescribed to ensure uniqueness.
• The method generalizes to higher codimensions, extending the solution beyond $\mathbb{S}^3$ to $\mathbb{S}^n$ for $n \geq 3$.
• The construction is invariant under Möbius transformations, confirming the conformal naturality of the solution.