Geodesic
Natural pseudodistances between closed surfaces
Journal of the European Mathematical Society, Tome 9 (2007) no. 2, pp. 331-353.

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Let us consider two closed surfaces M, N of class C1 and two functions φ:M→R, ψ:N→R of class C1, called measuring functions. The natural pseudodistance d between the pairs (M,φ), (N,ψ) is defined as the infimum of Θ(f)=defmaxP∈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 ∣c1​−c2​∣ or 21​∣c1​−c2​∣, or 31​∣c1​−c2​∣, where c1​ and c2​ 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.
DOI : 10.4171/jems/82
Classification : 58-XX, 49-XX, 53-XX, 00-XX
Keywords: Natural pseudodistance, measuring function, harmonic map 
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Pietro Donatini; Patrizio Frosini. Natural pseudodistances between closed surfaces. Journal of the European Mathematical Society, Tome 9 (2007) no. 2, pp. 331-353. doi : 10.4171/jems/82. http://geodesic.mathdoc.fr/articles/10.4171/jems/82/

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