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

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

### Citations:

Zbl 0634.53045; Zbl 0122.401
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