Minimal surfaces with constant curvature in 4-dimensional space forms

We classify minimal surfaces with constant Gaussian curvature in a 4 4 -dimensional space form without any global assumption. As a corollary of the main theorem, we show there is no isometric minimal immersion of a surface with constant negative Gaussian curvature into the unit 4 4 -sphere even locally. This gives a partial answer to a problem proposed by S. T. Yau.