JournalsjemsVol. 9 , No. 2DOI 10.4171/jems/82

Natural pseudodistances between closed surfaces

  • Patrizio Frosini

    Università di Bologna, Italy
  • Pietro Donatini

    Università di Bologna, Italy
Natural pseudodistances between closed surfaces cover


Let us consider two closed surfaces M\mathcal{M}, N\mathcal{N} of class C1C^1 and two functions φ:MR\varphi:{\mathcal{M}}\rightarrow \mathbb{R}, ψ:NR\psi:\mathcal{N}\rightarrow \mathbb{R} of class C1C^1, called measuring functions. The natural pseudodistance d{d} between the pairs (M,φ)({\mathcal{M}},\varphi), (N,ψ)({\mathcal{N}},\psi) is defined as the infimum of Θ(f)=defmaxPMφ(P)ψ(f(P))\Theta(f)\stackrel{def}{=}\max_{P\in \mathcal{M}}|\varphi(P)-\psi(f(P))|, as ff varies in the set of all homeomorphisms from M\mathcal{M} onto N\mathcal{N}. In this paper we prove that the natural pseudodistance equals either c1c2|c_1-c_2| or 12c1c2\frac{1}{2}|c_1-c_2|, or 13c1c2\frac{1}{3}|c_1-c_2|, where c1c_1 and c2c_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.