Geometric interpretation of energy-momentum tensor and Lagrangian associated to a soliton equation

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

• Are you asking for the geometric interpretations of the Lagrangians of the Gauss-Codazzi-Mainardi equation and the sinh-Gordon equation or are you asking whether there is some (common) geometric interpretation of all Lagrangian functions generating equations stemming from the above immersion and the demand that the first fundamental form is diagonal? – Konstantinos Kanakoglou Mar 22 '17 at 17:52
• I am asking whether there exist, for a general immersion of a 2D surface into 3D space with diagonal metric, a geometric meaning for the Lagrangian generating the Gauss-Codazzi-Mainardi equation. I'd also like to understand the geometric role of the associated energy-momentum tensor. Sorry if that was not clear – Stefano Mar 22 '17 at 19:33