Natural pseudodistances between closed surfaces

Let us consider two closed surfaces M, N of class C^1 and two functions phi:M-->R, psi:N-->R of class C^1, called measuring functions. The natural pseudodistance d between the pairs (M,phi), (N,psi) is defined as the infimum of Theta(f)=max_M |phi-psi ° f|, as f varies in the set of all homeomorphisms from M onto N. In this paper we prove that the natural pseudodistance equals either |c_1-c_2| or 1/2|c_1-c_2|, or 1/3|c_1-c_2|, where c_1 and c_2 are two suitable critical values of the measuring functions. This equality shows that a previous relation between natural pseudodistance and critical values obtained in general dimension can be improved in the case of closed surfaces. Our result is based on a theorem by Jost and Schoen concerning harmonic maps between surfaces.