The class of second fundamental forms arising from minimal immersions in a space form

The second fundamental form arising from an oriented minimal immersion of a closed surface in a space form satisfies several constraints.One of them is provided by the Gauss-Codazzi equation that can be rephrased as a semilinear problem on the surface.We discuss some results for these type of nonlinear problems and analyze the behaviors of the solutions when the hyperbolic norm of the second fundamental form is small.