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Perron's Method for General Quasilinear Symmetric Hyperbolic Systems and Immortal C 1 Viscosity Solutions for the Einstein Cauchy Problem

We extend all previous results of our [Sm4], [Sm5] to the case of general (non-diagonal coefficents) Quasilinear First Order Hyperbolic Systems, in slab domains with an odd number of Spacial Dimensions greater than or equal to three, prove a comparision Principle for viscosity sub and super solutions of such general systems, and apply our results to the Cauchy Problem for the Einstein Field Equations. We show–if the initial data, not required to be small, satisfies a barrier condition, and slightly weaker regularity than required in [Mf] –that unique C viscosity solutions of the Einstein Cauchy Problem exist for all time.