HARMONIC MORPHISMS FROM EINSTEIN 4-MANIFOLDS TO RIEMANN SURFACES

• DOI: 10.1142/s0129167x0300179x
• Publication Date: 2003-05-26T08:20:29Z
• Authors: MARINA VILLE
• Keywords: 58E20; Superminimal; Harmonic maps; Real analyticity; 53C42; Hermitian–Einstein; 01 natural sciences; 510; MSC : 53C25 ; 53C42 ; 58E20 ; 32L07; MSC : 53C25; [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG]; 0101 mathematics; 32L07

If M and N are Riemannian manifolds, a harmonic morphism f : M → N is a map which pulls back local harmonic functions on N to local harmonic functions on M. If M is an Einstein 4-manifold and N is a Riemann surface, John Wood showed that such an f is holomorphic w.r.t. some integrable complex Hermitian structure defined on M away from the singular points of f. In this paper we extend this complex structure to the entire manifold M. It follows that there are no non-constant harmonic morphisms from [Formula: see text] or [Formula: see text] to a Riemann surface. The proof relies heavily on the real analyticity of the whole situation. We conclude by an example of a non-constant harmonic morphism from [Formula: see text] to [Formula: see text].