We study the timelike minimal surface problem in spatially homogeneous and isotropic expanding spacetimes. Such "surfaces", which are in fact immersed submanifolds, are stationary with respect to variations of the induced volume functional and can by described by solutions to a non-linear coupled system of hyperbolic partial differential equations. We study the Cauchy problem associated with these equations in the case where the ambient spacetime is spatially flat and obtain a local well-posedness result, provided that the set of initial data is, in some appropriate sense, sufficiently small. Our well-posedness result in accompanied by an "extension criterion" which we use to show small-data global existence of critical timelike submanifolds in the Minkowski and de Sitter ambient spacetimes.
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