Can Geodesic Surfaces Minimize Area Like Curves Minimize Length?

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SUMMARY

Geodesic surfaces can indeed minimize area analogous to how geodesic curves minimize length. The discussion references Plateau's problem in Euclidean space, which addresses the minimization of surface area. In Minkowski space, the Nambu-Goto action is mentioned as a framework for maximizing area in the context of space-time surfaces. Theoretical exploration of these concepts can deepen understanding of geometric properties in various mathematical contexts.

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  • Understanding of geodesic curves and their properties
  • Familiarity with Plateau's problem in Euclidean geometry
  • Basic knowledge of Minkowski space and its implications in physics
  • Concept of the Nambu-Goto action in theoretical physics
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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 ?,
 
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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".
 
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http://math.rice.edu/~polking/Math410/ lists some examples.
 
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