二维球面在酉群中的常曲率极小浸入

中文摘要 AI 辅助

本文研究了二维球面 $S^2$ 到酉群 $\mathrm{U}(n)$ 的常曲率极小浸入的刚性结果。利用调和映射的圈群方法，我们建立了这类浸入与 $S^2$ 到有限维格拉斯曼流形的一类特殊常曲率全纯浸入之间的对应关系。在 $\mathrm{U}(3)$ 的情形下，我们分类了单位子数为一的常曲率极小浸入，并证明了在自然无分歧条件下，单位子数为二的此类浸入是 $S^1$-不变的；作为推论，每个单位子数为二的完全无分歧常曲率极小浸入 $S^2\to \mathrm{U}(3)$ 都酉等价于 $\mathbb{C}P^2$ 中 Veronese 曲线的第一高斯映射与 Cartan 嵌入 $\mathbb{C}P^2\hookrightarrow \mathrm{U}(3)$ 的复合。

英文摘要

In this article, we investigate rigidity results for constantly curved minimal immersions of the two-sphere $S^2$ into the unitary group $\mathrm{U}(n)$. Using loop group methods for harmonic maps, we establish a correspondence between such immersions and a distinguished class of constantly curved holomorphic immersions of $S^2$ into finite-dimensional Grassmannians. In the case $\mathrm{U}(3)$, we classify the constantly curved minimal immersions of $S^2$ with uniton number one and prove that, under a natural unramifiedness condition, those of uniton number two are $S^1$-invariant; as a consequence, every constantly curved totally unramified minimal immersion $S^2\to \mathrm{U}(3)$ of uniton number two is unitarily congruent to the composition of the first Gauss map of the Veronese curve in $\mathbb{C}P^2$ with the Cartan embedding $\mathbb{C}P^2\hookrightarrow \mathrm{U}(3)$.

发表机构

• Universidade da Beira Interior（比拉内乌大学）

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