Integer-valued characteristics of solutions of the noncommutative sigma model

Any finite-energy solution of a noncommutative sigma model has three nonnegative integer-valued characteristics: the normalized energy $e(\Phi)$, canonical rank $r(\Phi)$, and minimum uniton number $u(\Phi)$. We prove that $r(\Phi)\ge u(\Phi)$ and $e(\Phi)\ge u(\Phi)(u(\Phi)+1)/2$. Given any numbers $e,r,u\in\mathbb N$ that satisfy the slightly stronger inequalities $r\ge u$ and $e\ge r+u(u-1)/2$, we construct a finite-energy solution $\Phi$ with $e(\Phi)=e$, $r(\Phi)=r$, and $u(\Phi)=u$.