Define a complex valued function $z$ in the stereographic plane and let $z=z(s,s^{*})$ for $s$ a complex valued variable and $*$ denoting the complex conjugate.
Define the Dirichlet energy as
$$ \int \frac{\lvert z_s \rvert^{2} + \lvert z_{s^{*}} \rvert^{2}}{(1 + \lvert z \rvert^{2})^{2}} \ ds ds^*. $$
Minimizing this with respect to $z^*$ we find that $z$ must satisfy the following nonlinear PDE:
$$(1 + \lvert z \rvert^{2}) z_{s s^*} = 2 z^{*} z_s z_{s^{*}}. $$
Some simple solutions to this are $z = f(s)$ or $z = g(s^{*})$ for arbitrary functions $f, g$.
Are there other known solutions to this equation?
Edit added: Are these solutions related to the spherical harmonics? If so, how can we compose two solutions in the stereographic plane to find another solution?
I see reference to this equation here, but cannot find other useful references. Any tips on work that has been done on this equation are appreciated.
