I have a question for you.

BACKGROUND

Consider an immersion $F\,:\;\mathscr S\;\longrightarrow\;\mathbb E^3$ of a surface $\mathscr S$ in the $3$-D euclidean plane $\mathbb E^3$ with canonical scalar product $\langle\cdot ,\cdot \rangle$. It is possible to find a metric structure such that the first fundamental form is diagonal $$\langle dF,dF\rangle=4e^udzd\bar z\;,\qquad dF=F_zdz+F_{\bar z}d\bar z\;.$$ It is well known that, in the case $\mathscr S$ is a Constant Mean Curvature surface, the field $u$ is constrained by the modified sinh-Gordon equation $$u_{z\bar z}+\frac{1}{2}\left(e^{u}+A\bar Ae^{-u}\right)=0\;,\qquad A_{\bar z}=\bar A_z=0\;,$$ which are simply the Gauss-Codazzi-Mainardi equation. If, on the other hand, we choose $\mathscr S$ to be a minimal surface we would get the Liouville equation $$u_{z\bar z}+\frac{\kappa}{2}e^{-u}=0\;,\qquad \kappa=\textrm{const.}\;.$$

QUESTION

I would like to know if there exists a geometric interpretation for the Lagrangian $L$ and the Energy-Momentum tensor $T$ associated to the nonlinear field equation for $u$ (for example the sinh-Gordon equation above). I could not find a clear reference and would be glad if someone can suggest some.

Thanks a lot!

Stefano