球面中任意亏格极小曲面的高维族
High-Dimensional Families of Minimal Surfaces of Arbitrary Genus in Round Spheres
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中文总结 AI 辅助
本文在偶数维球面中构造任意亏格与共形结构的极小曲面的任意高维参数族，其维数随次数趋于无穷。
中文摘要 AI 辅助
我们证明了在偶数维球面中存在任意指定亏格与共形结构的极小曲面的任意高维族。更精确地，设$n\geq2$且$\Sigma$为亏格$g$的任意闭黎曼曲面。我们构造一个次数序列$d_\ell\to+\infty$，使得对每个$\ell$，存在一个复维数为$2d_\ell+n^2(1-g)$的复流形，由$\Sigma$到$\mathbb S^{2n}$的度为$d_\ell$的线性满秩分支超极小浸入组成。特别地，这产生了线性满秩分支极小浸入的参数化族，其维数趋于无穷。
英文摘要
We prove the existence of arbitrarily high-dimensional families of minimal surfaces of any prescribed genus and conformal structure in even-dimensional round spheres. More precisely, let $n\geq2$ and let $Σ$ be any closed Riemann surface of genus $g$. We construct a sequence of degrees $d_\ell\to+\infty$ such that, for every $\ell$, there exists a complex manifold of complex dimension $2d_\ell+n^2(1-g)$ consisting of linearly full branched superminimal immersions of $Σ$ into $\mathbb S^{2n}$ of degree $d_\ell$. In particular, this yields parametrized families of linearly full branched minimal immersions whose dimensions tend to infinity.
发表机构
- Mathematics Institute, University of Warwick（华威大学数学研究所）
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