Quasiconformal harmonic maps into negatively curved manifolds

Source abstract

Let $F:M\to N$ be a harmonic map between complete Riemannian manifolds. Assume that $N$ is simply connected with sectional curvature bounded between two negative constants. If $F$ is a quasiconformal harmonic diffeomorphism, then $M$ supports an infinite dimensional space of bounded harmonic functions. On the other hand, if $M$ supports no non-constant bounded harmonic functions, then any harmonic map of bounded dilation is constant.