Natural pseudodistances between closed surfaces

Let us consider two closed surfaces $\mathcal{M}$, $\mathcal{N}$ of class $C^1$ and two functions $\varphi:{\mathcal{M}}\rightarrow \mathbb{R}$, $\psi:\mathcal{N}\rightarrow \mathbb{R}$ of class $C^1$, called measuring functions. The natural pseudodistance ${d}$ between the pairs $({\mathcal{M}},\varphi)$, $({\mathcal{N}},\psi)$ is defined as the infimum of $\Theta(f)\stackrel{def}{=}\max_{P\in \mathcal{M}}|\varphi(P)-\psi(f(P))|$, as $f$ varies in the set of all homeomorphisms from $\mathcal{M}$ onto $\mathcal{N}$. In this paper we prove that the natural pseudodistance equals either $|c_1-c_2|$ or $\frac{1}{2}|c_1-c_2|$, or $\frac{1}{3}|c_1-c_2|$, 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.