Constantly curved minimal immersions of the two-sphere in unitary groups

Abstract

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 $\mathbb{C}{P}^2\hookrightarrow \mathrm{U}(3)$.