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 ${\Sigma }_1$ and ${\Sigma }_2$ are surfaces with conformal metrics ${\sigma }^{2}dzd\bar{z}$ and ${\rho }^{2}dud\bar{u}$, resp., and $u:{\Sigma }_1\to {\Sigma }_2$ is harmonic, then

is a holomorphic quadratic differential on ${\Sigma }_1$ (and $\varphi$ 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 $\varphi$ is harmonic.

In the article under consideration, a variational procedure is invented that produces a map with holomorphic $\varphi$ 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.

Résumé

Dans cette note, on étudie la relation entre deux définitions pour les applications harmoniques $u$ en dimension deux, l’une étant que la forme différentielle quadratique

(associée à $u$, où $z=x+iy$ est une coordonnée conforme locale) soit holomorphe, l’autre étant l’équation différentielle du deuxième ordre

(2) implique que $\omega$ soit holomorphe. Nous déterminons, dans quelle mesure l’implication inverse n’est pas vraie, et donnons une réponse négative à une question de Eells–Lemaire. De plus, nous construisons une procédure variationnelle, qui donne des revêtements ramifiés conformes de $S^2$.