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Suppose $M$ and $N$ are Riemannian manifolds (non compact) of dimension $2$ and $f$ is an harmonic map between $M$ and $N$. When is $f$ conformal?

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You may like to check the series in Contemporary Mathematics, Volume 308, Differential Geometry and Integrable Systems: a conference on integrable systems in differential geometry, University of Tokyo, Japan, July 17-21, 2000; edited by M. Guest, R. Miyaoko, Y. Ohnito.

It has a host of result that might interest you. For example,

If $M$ is a non-compact Riemannian manifold with $Ricci(M)\geq0$ and $Ricci(N)\leq0$ then any harmonic map with finite energy is a constant.

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