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$\newcommand{\TM}{\operatorname{TM}}$ $\newcommand{\M}{\mathcal{M}}$ $\newcommand{\N}{\mathcal{N}}$ $\newcommand{\TM}{\operatorname{T\M}}$ $\newcommand{\TN}{\operatorname{T\N}}$ $\newcommand{\TstarM}{...

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?

By the uniformization theorem, for every genus-0 closed surface $\mathcal{M}\subset\mathbb{R}^3$, there is a conformal map $f:\mathcal{M}\rightarrow \mathbb{S}^2$. Furthermore consider the Dirichlet ...

Edit according to the comment of user36931 I remove the "motivation" from the previous version and I add an statement to the first question We consider the following two classes of smooth maps on $...