A conformal representation for linear Weingarten surfaces in the de Sitter space

In this paper we study a wide family of spacelike surfaces in S13 which includes, for instance, constant mean curvature 1 surfaces and flat surfaces: those whose mean and Gauss–Kronecker curvatures verify the lineal relationship 2a(H−1)+b(K−1)=02a(H−1)+b(K−1)=0 for a,b∈R,a+b≠0a,b∈R,a+b≠0. We show that these surfaces can be parametrized with holomorphic data and we use it to classify the complete surfaces from this family with non-negative Gaussian curvature.