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arXiv 0903.4836math.DGmath.AG
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$S^3$中的高亏格极小曲面与稳定丛

Higher genus minimal surfaces in $S^3$ and stable bundles

Sebastian Heller

发表信息：
J. Reine Angew. Math. 685 (2013)
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摘要：

我们考虑亏格为2的紧致极小曲面$f\colon M\to S^3$，它与嵌入同伦.我们假设伴随的全纯丛是稳定的。我们证明了这些表面可以构造从一个整体定义的家庭亚纯联络的DPW方法。亚纯联络的极点在阶数不超过2的黎曼曲面的Weierstrass点上。为了证明DPW势的存在性，我们利用$P H^0(M;K^2).$的对偶$S^{-1}$，给出了自旋丛$S$的稳定扩张$0\to S^{-1}\to V\to S\to 0$的一个特征.我们还考虑了与$S^3.$中极小曲面相关联的全纯结构族.对于亏格$g\geq2$的曲面，其联络的完整性一般是非阿贝尔的，因此全纯结构是一般稳定的.

英文摘要：

We consider compact minimal surfaces $f\colon M\to S^3$ of genus 2 which are homotopic to an embedding. We assume that the associated holomorphic bundle is stable. We prove that these surfaces can be constructed from a globally defined family of meromorphic connections by the DPW method. The poles of the meromorphic connections are at the Weierstrass points of the Riemann surface of order at most 2. For the existence proof of the DPW potential we give a characterization of stable extensions $0\to S^{-1}\to V\to S\to 0$ of spin bundles $S$ by its dual $S^{-1}$ in terms of an associated element of $P H^0(M;K^2).$ We also consider the family of holomorphic structures associated to a minimal surface in $S^3.$ For surfaces of genus $g\geq2$ the holonomy of the connections is generically non-abelian and therefore the holomorphic structures are generically stable.

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