Can Geodesic Surfaces Minimize Area Like Curves Minimize Length?

[mhill]

if we have or can have geodesic curves minimizing the integral $$\sqrt (g_{ab}\dot x_a \dot x_b )$$ is there a theory of 'minimizing surfaces or Geodesic surfaces' that minimize the Area or a surface ?,

 

[gel]

I don't know about the general theory, but in Euclidean space this is http://en.wikipedia.org/wiki/Plateau%27s_problem" .
For space-time surfaces, in Minkowski space, it is similar to the simplest (non-quantized) http://en.wikipedia.org/wiki/Nambu-Goto_action" except I assume you would be maximizing the "area".

 

[robphy]

http://math.rice.edu/~polking/Math410/ lists some examples.