# Natural pseudodistances between closed surfaces

### Patrizio Frosini

Università di Bologna, Italy### Pietro Donatini

Università di Bologna, Italy

## Abstract

Let us consider two closed surfaces $M$, $N$ of class $C_{1}$ and two functions $φ:M→R$, $ψ:N→R$ of class $C_{1}$, called measuring functions. The natural pseudodistance $d$ between the pairs $(M,φ)$, $(N,ψ)$ is defined as the infimum of $Θ(f)=defmax_{P∈M}∣φ(P)−ψ(f(P))∣$, 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 $21 ∣c_{1}−c_{2}∣$, or $31 ∣c_{1}−c_{2}∣$, 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.

## Cite this article

Patrizio Frosini, Pietro Donatini, Natural pseudodistances between closed surfaces. J. Eur. Math. Soc. 9 (2007), no. 2, pp. 331–353

DOI 10.4171/JEMS/82