  We show existence and uniqueness for timelike minimal submanifolds (world
volume of p-branes) in ambient Lorentz manifolds admitting a time function in a
neighborhood of the initial submanifold. The initial value formulation
introduced and the conditions imposed on the initial data are given in purely
geometric terms. Only an initial direction must be prescribed in order to
provide uniqueness for the geometric problem. The result covers non-compact
initial submanifolds of any codimension. By considering the angle of the
initial direction and vector fields normal to the initial submanifold with the
unit normal to the foliation given by the time function we obtain a
quantitative description of "distance" to the light cone. This description
makes it possible to treat initial data which are arbitrarily close to the
light cone. Imposing uniform assumptions give a lower bound for a notion of
"time of existence" depending only on geometric quantities involving the length
of timelike curves lying in the solution.
