Weierstrass Type Representation of Willmore Surfaces in S n

In this paper, we reformulate the EulerLagrange equations of Willmore surfaces in S n as the flatness of a family of certain loop algebravalued 1forms. Therefore we can give the Weierstrass type representation of conformal Willmore surfaces. We also discuss the relations between conformal Willmore surfaces in S n and minimal surfaces in constant curvature spaces S n, R n, H n, and prove that some special Willmore surfaces can be derived from minimal surfaces in S n, R n, H n.