Natural pseudodistances between closed surfaces

Abstract

Let us consider two closed surfaces M, N of class C1 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) := maxP∈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|, 1/2 |c1 - c2|, or 1/3|c1 - c2|, where c1 and c2 are two suitable critical values of the measuring functions. This shows that a previous relation between the 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. © European Mathematical Society 2007.