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Natural pseudodistances between closed surfaces

Pietro Donatini, Patrizio Frosini (2007)

Journal of the European Mathematical Society

Let us consider two closed surfaces , 𝒩 of class C 1 and two functions ϕ : , ψ : 𝒩 of class C 1 , called measuring functions. The natural pseudodistance d between the pairs ( , ) , ( 𝒩 , ψ ) is defined as the infimum of Θ ( f ) : = max P | ϕ ( P ) ψ ( f ( P ) ) | as f varies in the set of all homeomorphisms from onto 𝒩 . In this paper we prove that the natural pseudodistance equals either | c 1 c 2 | , 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 shows that a previous relation between the natural pseudodistance and critical values...

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