A note on harmonic maps between surfaces

  • Jürgen Jost

    Centre for Mathematical Analysis, Hanna Neumann Building, Australian National University, GPO Box 4, Canberra ACT 2601, Australia

Abstract

In this note, the relation between harmonic maps between surfaces and holomorphic quadratic differentials is investigated. Remember that if and are surfaces with conformal metrics and , resp., and is harmonic, then

is a holomorphic quadratic differential on (and vanishes identically if and only if is conformal).

It has been an open question to which extent the converse is true, i.e. whether a map with holomorphic is harmonic.

In the article under consideration, a variational procedure is invented that produces a map with holomorphic 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 en dimension deux, l’une étant que la forme différentielle quadratique

(associée à , est une coordonnée conforme locale) soit holomorphe, l’autre étant l’équation différentielle du deuxième ordre

(2) implique que 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 .

Cite this article

Jürgen Jost, A note on harmonic maps between surfaces. Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (1985), no. 6, pp. 397–405

DOI 10.1016/S0294-1449(16)30393-6