Local existence of quasispherical space-time initial data

We consider the system of Einstein constraint equations in the special case when the spacelike 3-manifold is assumed to satisfy the quasispherical ansatz of Bartnik. We introduce two different time slicing conditions, under each of which we may view the resulting constraint equations as a parabolicellipticODE system of partial differential equations. We combine recent existence results for parabolic equations in Sobolev space with an iterative method to prove local existence results for the quasispherical Einstein constraint equations under the two different time slicing conditions.