On the Bj\"orling problem for Willmore surfaces
On the Bj\"orling problem for Willmore surfaces
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DOI：
10.4310/jdg/1519959622
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发表时间：
2014-09
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影响因子：
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通讯作者：
D. Brander;Pengfu Wang
中科院分区：
文献类型：
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作者：
D. Brander;Pengfu Wang
我们通过调和映射表示解决了Willmore曲面的Bj\“orling问题的类似问题。对于无脐带缆的情况，问题和解决方案如下：给定S^3 $中的一条真实的解析曲线y_0 $，连同曲面法线和沿曲线的对偶Willmore曲面沿着的值的规定，提升到Minkowski 5 $-空间R\mathbb{R}^5_1$中的光锥，我们证明，利用各向同性调和映射，存在唯一的一对对偶Willmore曲面$y$和$\hat y$满足沿曲线沿着的给定值。我们给出了曲面对的广义Weierstrass数据的显式公式。对于三维目标，我们使用的解决方案来明确地描述维尔斯特拉斯数据，在几何量方面，所有的等变Willmore表面。对于曲面有脐点的情况，我们应用H\'{e}lein引入的更一般的半各向同性调和映射来导出一个解：在这种情况下，映射$\hat y$不一定是对偶曲面，并且必须规定$\hat y$的导数的附加数据。这个解决方案是广义的更高的余维。
We solve the analogue of Bj\"orling'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 H\'{e}lein 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.