A geometric characterization of harmonic diffeomorphisms between surfaces
Let (M,g) and (N,h) denote oriented Riemann surfaces with metrics g and h. A quadratic differential \(Q_ N\) on N is said to be \(\rho\)- holomorphic if, away from its zeros, \(Q_ N\) takes the form \((\omega +i*\rho \omega)^ 2\) where \(\omega\) and *\(\rho\omega\) are closed 1-forms and \(\rho(y)=(1-4| \omega |^ 2_ y)^{-1/2}\) for \(y\in N\). We prove the following: For every nonholomorphic, orientation preserving harmonic diffeomorphism f: (M,g)\(\to(N,h)\), there exists a canonical pair of orthogonal measured foliations, \(F_ 1\) and \(F_ 2\), associated to a holomorphic quadratic differential \(Q_ M=(f*h)^{2,0}\) on M, such that \(f(F_ 1)\) and \(f(F_ 2)\) are a pair of orthogonal measured foliations, associated to a \(\rho\)-holomorphic quadratic differential \(Q_ N=(f^{- 1})^*Q_ M\) on N. Let (N,h) be compact and real analytic. There exists a one-to-one correspondence between the set of real analytic, \(\rho\)- holomorphic quadratic differentials on N and the set of harmonic diffeomorphisms into N determined up to composition with a conformal diffeomorphism.
scientific article; zbMATH DE number 3851954