Ergodicity of the geodesic flow on surfaces of nonpositive curvature

We give a full proof that the geodesic flow of a closed surface of genus at least two with a $C^{3+\mathrm{L}\mathrm{i}\mathrm{p}}$ metric of nonpositive curvature is ergodic with respect to Liouville measure. In a forthcoming paper, we will present the corresponding result for smooth rank-one manifolds in every dimension.