Gauss maps of surfaces. (English) Zbl 0581.58013

Eells, James

Perspectives in mathematics, Anniv. Oberwolfach 1984, 111-129 (1984).

[For the entire collection see Zbl 0548.00010.]
This is a comprehensive account on what is known at present about the theory of minimal branched immersions (conformal harmonic maps) \(\phi\) : \(M\to N\) of a Riemann surface M into a Riemannian manifold N. Such maps \(\phi\) are studied via their associated Gauss maps \(\gamma_{\phi}\) of M into the respective complex quadric \(Q_{n-2}\); the key results being the theorem of Chern (\(\phi\) is harmonic iff \(\gamma_{\phi}\) is anti- holomorphic) and that of Ruh-Vilms (\(\phi\) has constant mean curvature iff \(\gamma_{\phi}\) is harmonic). One of the goals is to obtain a parametrization of the conformal harmonic maps \(\phi\) : \(M\to N\) in terms of (partially) holomorphic maps of M into a suitable twistor space over N, a result which is motivated by a theorem of Calabi on isotropic harmonic maps into real projective spaces. The paper is concluded with a collection of various unsolved problems.

Reviewer: G.Toth

MSC:

58E20

Harmonic maps, etc.

Keywords:

conformal map; Riemann surface; harmonic maps

Citations:

Zbl 0548.00010