Geodesic surface

[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 Plateau's problem (http://en.wikipedia.org/wiki/Plateau%27s_problem).
For space-time surfaces, in Minkowski space, it is similar to the simplest (non-quantized) string theory (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.