On a stack of surfaces obtained from the $\mathbb{C}P^{N-1}$ sigma models

Under the assumption that the $\mathbb{C}P^{N-1}$ sigma model is defined on the Riemann sphere and its action functional is finite, we derive surfaces induced by surfaces and we demonstrate that the stacked surfaces coincide with each other, which means idempotency of the recurrent procedure. Along the path to the solutions of the $\mathbb{C}P^{N-1}$ model equations, we demonstrate that the Euler--Lagrange equations for the projectors admit larger classes of solutions than the ones corresponding to rank-1 projectors.