Let us consider two closed surfaces , of class and two functions , of class , called measuring functions. The natural pseudodistance between the pairs , is defined as the infimum of , as varies in the set of all homeomorphisms from onto . In this paper we prove that the natural pseudodistance equals either or , or , where and 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. 231–253DOI 10.4171/JEMS/82