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In this paper, we consider both differential and algebraic properties ofsurfaces associated with sigma models. It is shown that surfaces defined by thegeneralized Weierstrass formula for immersion for solutions of the CP^{N-1}sigma model with finite action, defined in the Riemann sphere, are themselvessolutions of the Euler-Lagrange equations for sigma models. On the other hand,we show that the Euler-Lagrange equations for surfaces immersed in the Liealgebra su(N), with conformal coordinates, that are extremals of the areafunctional subject to a fixed polynomial identity are exactly theEuler-Lagrange equations for sigma models. In addition to these differentialconstraints, the algebraic constraints, in the form of eigenvalues of theimmersion functions, are treated systematically. The spectrum of the immersionfunctions, for different dimensions of the model, as well as its symmetryproperties and its transformation under the action of the ladder operators arediscussed. Another approach to the dynamics is given, i.e. description in termsof the unitary matrix which diagonalizes both the immersion functions and theprojectors constituting the model.

Soliton surfaces associated with sigma models; differential and algebraic aspect