an:03949221
Zbl 0591.53031
Salamon, Simon
Harmonic and holomorphic maps
EN
Geometry Semin. ''Luigi Bianchi'', Lect. Sc. Norm. Super., Pisa 1984, Lect. Notes Math. 1164, 161-224 (1985).
1985
a
53C15 58E20 32Q99 53C55 53C20
harmonic maps; holomorphic maps; almost complex manifolds; twistor spaces; minimal surfaces; holomorphic curves; symmetric space; complex projective spaces; quaternionic K??hler manifolds; Gauss maps; CR manifolds
[For the entire collection see Zbl 0568.00011.]
This article is based on a series of seminars given at the Scuola Normale which relate the theories of harmonic maps between Riemannian manifolds and holomorphic maps between almost complex manifolds using twistor spaces. The author begins by explaining the result, due to Lichnerowicz, that a holomorphic map \(\phi\) : \(M\to N\) between almost Hermitian manifolds is harmonic if M is cosymplectic (i.e. \(d^*\omega_ M=0\) where \(\omega_ M\) is the fundamental 2-form) and N is (1,2)-symplectic (i.e. the (1,2)-component of \(d\omega_ N\) is zero). This is generalized by allowing N to be any oriented even dimensional Riemannian manifold with twistor space \(\pi\) : \(S\to N\). There is a non-integrable almost complex structure on S such that if \(\psi\) : \(M\to S\) is holomorphic then \(\pi\) \(\circ \psi\) is harmonic. Consideration of the case when M is a Riemann surface leads to a characterization of minimal surfaces in N in terms of holomorphic curves in S (in particular when N is a sphere). When N is a symmetric space S can be replaced by subbundles which are twistor spaces in a generalized sense. These techniques are used to prove results due to several authors on harmonic maps into complex projective spaces, quaternionic K??hler manifolds, Gauss maps, CR manifolds, minimal surfaces in \(S^ 4\) and holomorphic curves in \(S^ 6\).
F.Kirwan
Zbl 0568.00011