Let us consider two closed surfaces M, N of class C^1 and two functions phi:M-->R, psi:N-->R of class C^1, called measuring functions. The natural pseudodistance d between the pairs (M,phi), (N,psi) is defined as the infimum of Theta(f)=max_M |phi-psi ° f|, 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 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 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.

P. Donatini, P. Frosini (2007). Natural pseudodistances between closed surfaces. JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY, 9, 231-253 [10.4171/JEMS/82].

Natural pseudodistances between closed surfaces

DONATINI, PIETRO;FROSINI, PATRIZIO
2007

Abstract

Let us consider two closed surfaces M, N of class C^1 and two functions phi:M-->R, psi:N-->R of class C^1, called measuring functions. The natural pseudodistance d between the pairs (M,phi), (N,psi) is defined as the infimum of Theta(f)=max_M |phi-psi ° f|, 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 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 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.
2007
P. Donatini, P. Frosini (2007). Natural pseudodistances between closed surfaces. JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY, 9, 231-253 [10.4171/JEMS/82].
P. Donatini; P. Frosini
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11585/34787
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