A note on harmonic maps between surfaces

Abstract

In this note, the relation between harmonic maps between surfaces and holomorphic quadratic differentials is investigated. Remember that if Σ1 and Σ2 are surfaces with conformal metrics σ2dzdz¯ and ρ2dudu¯, resp., and u : Σ1 → Σ2 is harmonic, thenϕ:=ρ2uzu¯zdz2is a holomorphic quadratic differential on Σ1 (and ϕ vanishes identically if and only if u is conformal).It has been an open question to which extent the converse is true, i.e. whether a map with holomorphic ϕ is harmonic.In the article under consideration, a variational procedure is invented that produces a map with holomorphic ϕ in every homotopy class of maps between closed surfaces. While on one hand, thus conformal selfmaps of the two-sphere are obtained by a variational method, answering a question of Uhlenbeck, contrasting this existence result on the other hand with some nonexistence results for harmonic maps, one is led to a negative answer to the above converse question. An explicit example is displayed as well.