The construction of a harmonic extension of a map between the ideal boundaries of two complete noncompact manifolds of strictly negative curvature involves two parts: (i) the construction of an extension of the boundary map to the whole space with bounds on the energy density and the tension fields, (ii) deformation of the extension in (i) to a harmonic map.

I will describe recent progress due to Fotiadis, Markovic and myself concerning step (ii) which enables one to allow the boundary map to be considerably more singular than previously allowed by Li-Tam and others. I will also describe a proof by Fotiadis, different from the Teichmuller proof of Markovic, of the existence of a harmonic diffeomorphism of the hyperbolic plane which extends a symmetric map of the ideal circle.

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