Minimal Surfaces, Differential Geometry, and Partial Differential Equations

[octohydra]

Last night in a lecture my professor explained that some partial differential equations are used to observe events on minimal surface (e.g. membranes).

A former advisor, someone that studied differential geometry, gave a brief summary of minimal surfaces but in a diffy G perspective.

1.) Are there any connections between the two studies and their perspectives on minimal surfaces?
2.) Are there any papers that use minimal surfaces, PDE's, and Diffy G together in some manner?

I'm curious to know what do PDE's and Diffy G have in 'common' via minimal surfaces. Any input is appreciated.

 

[fresh_42]

Minimal surfaces are the critical points of the area functional $A(x)=\int \sqrt{g(u)}\,d^nu$ with $g(u)=(\det g_{ij}(u))$ and $g_{ij}(u)= \left( \dfrac{\partial x}{\partial u_i} \right)^\tau \dfrac{\partial x}{\partial u_j}\,.$