Non CMC Conformal Data Sets Which Do Not Produce Solutions of the Einstein Constraint Equations

Abstract

The conformal formulation provides a method for constructing and parametrizing solutions of the Einstein constraint equations by mapping freely chosen sets of conformal data to solutions, provided a certain set of coupled, elliptic determined PDEs (whose expression depends on the chosen conformal data) admit a unique solution. For constant mean curvature (CMC) data, it is known in almost all cases which sets of conformal data allow these PDEs to have solutions, and which do not. For non CMC data, much less is known. Here we exhibit the first class of non CMC data for which we can prove that no solutions exist.

To be published in the volume of Classical and Quantum Gravity in honour of Vincent Moncrief

References in corpus (4)

• Conformal ``thin sandwich'' data for the initial-value problem of general relativity
• Einstein Constraints on Asymptotically Euclidean Manifolds
• Crushing singularities in spacetimes with spherical, plane and hyperbolic symmetry
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Cited by in corpus (16)

• Rough Solutions of the Einstein Constraint Equations on Compact Manifolds
• Theorems on existence and global dynamics for the Einstein equations
• Rough solutions of the Einstein constraints on closed manifolds without near-CMC conditions
• Far-from-constant mean curvature solutions of Einstein's constraint equations with positive Yamabe metrics
• Near-Constant Mean Curvature Solutions of the Einstein Constraint Equations with Non-Negative Yamabe Metrics
• A model problem for conformal parameterizations of the Einstein constraint equations
• A limit equation associated to the solvability of the vacuum Einstein constraint equations using the conformal method
• Initial Value Problem in General Relativity
• The Conformal Method and the Conformal Thin-Sandwich Method Are the Same
• A new point of view on the solutions to the Einstein constraint equations with arbitrary mean curvature and small TT-tensor
• The Einstein-scalar field constraint system in the positive case
• The Einstein Constraint Equations on Compact Manifolds with Boundary
• The Constraint Equations in the Presence of a Scalar Field - the Case of the Conformal Method with Volumetric Drift
• A non-existence result for a generalization of the equations of the conformal method in general relativity
• Construction of vacuum initial data by the conformally covariant split system
• Einstein-type elliptic systems