The splitting and deformations of the generalized gauss map of compact CMC surfaces

We show that a non-conformal harmonic map from a Riemann surface into the Euclidean n-sphere can be considered as a component of minimal surfaces in higher dimensional spheres. In the same principle, we show that the generalized Gauss map of constant mean curvature surfaces in the 3-sphere globally splits into two non-conformal harmonic maps into the 2-sphere. Using this, we obtain examples of non-trivial harmonic map deformations for compact Riemann surfaces of arbitrary positive genus. In particular, we give a lower bound for the nullity (as harmonic maps) of the generalized Gauss map of compact CMC surfaces in the 3-sphere. Furthermore, we obtain an affirmative answer to Lawson’s conjecture for superconformal minimal surfaces in 4m-spheres.