Twisted harmonic maps and the self-duality equations. (English) Zbl 0634.53046

Donaldson, S. K.

Proc. Lond. Math. Soc., III. Ser. 55, 127-131 (1987).

Let \({\mathbb{H}}^ 3:=\{H\in M_{2\times 2}({\mathbb{C}})|\) \(H^*=H\), det H\(=1\}\) be the hyperbolic 3-space with constant negative curvature and \({\mathcal H}:=\tilde X\times_{\rho}{\mathbb{H}}^ 3\). Using ideas of J. Eells jun. and J. H. Sampson [Am. J. Math. 86, 109-160 (1964; Zbl 0122.401)] it is shown that for every compact Riemannian manifold X with universal cover \(\tilde X\) and irreducible representation \(\rho\) : \(\pi\) \({}_ 1(X)\to PSL(2,{\mathbb{C}})\) the flat \({\mathbb{H}}^ 3\) bundle \({\mathcal H}\to X\) has a harmonic section. (Here a section s is defined to be harmonic, if it is an extremum of a certain “energy” functional \(E(s)=\int_{X}| Ds|^ 2 d\mu.)\)
To each harmonic section there is associated a solution of Hitchin’s self-duality equation, hence this paper completes the circle of ideas introduced by N. J. Hitchin [ibid., 59-126 (1987; see the review above)].

Reviewer: S.Timmann

MSC:

53C55	Global differential geometry of Hermitian and Kählerian manifolds
53C20	Global Riemannian geometry, including pinching
58E20	Harmonic maps, etc.
30F15	Harmonic functions on Riemann surfaces

Keywords:

hyperbolic 3-space; harmonic section; self-duality equation