  In this paper, we study rank-1 projector solutions to the completely
integrable Euclidean $CP^{N-1}$ sigma model in two dimension and their
associated surfaces immersed in the $su(N)$ Lie algebra. We reinterpret and
generalize the proof of A.M. Din and W.J. Zakzrewski [1980] that any solution
for the $CP^{N-1}$$ sigma model defined on the Riemann sphere with finite
action can be written as a raising operator acting on a holomorphic one, or a
lowering operator acting on a antiholomorphic one. Our proof is formulated in
terms of rank-1 Hermitian projectors so it is explicitly gauge invariant and
gives new results on the structure of the corresponding sequence of rank-1
projectors. Next, we analyze surfaces associated with the $CP^{N-1}$ models
defined using the Generalized Weierstrass Formula for immersion, introduced by
B. Konopelchenko [1996]. We show that the surfaces are conformally
parameterized by the Lagrangian density with finite area equal to the action of
the model and express several other geometrical characteristics of the surface
in terms of the Lagrangian density and topological charge density of the model.
We demonstrate that any such surface must be orthogonal to the sequence of
projectors defined by repeated application of the raising and lowering
operators. Finally, we provide necessary and sufficient conditions that a
surface be related to a $CP^{N-1}$ sigma model.
