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arXiv · 2312.17706

Optimal localization for the Einstein constraints

Bruno Le Floch·Philippe G. LeFloch
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Abstract

We establish the existence of asymptotically Euclidean solutions, or initial data sets, to Einstein's vacuum constraints exhibiting both gravitational shielding and arbitrarily low decay. We resolve a conjecture of Carlotto and Schoen on gluing two solutions across an asymptotically conical domain: in the interpolation region we establish optimal estimates at and beyond harmonic decay, together with ADM invariant estimates. Starting from a seed data set that asymptotically solves the constraints, we project it to an exact solution, whose difference from the seed inherits the natural optimal radial decay rate. The seed-to-solution projection operator composes the linearized constraint operator with its formal adjoint. Its harmonic kernel at infinity generates energy-momentum modulators, which define relative invariants equal to the differences of the usual ADM invariants whenever those exist. We introduce harmonic, radial, and shell stability conditions on the angular gluing function; they ensure the positivity and radial monotonicity of the functionals governing optimal decay. A follow-up paper proves these conditions from weighted Poincar\'e, Korn, and Hardy inequalities, leading to verifiable conditions on the localization function. As particular cases, our theory applies to solutions without localization and for solutions in the small aperture limit. Our analysis of the linearized scalar curvature operator relies crucially on a new fourth-order curvature functional, and also applies to gluing scalar-flat metrics.

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Bruno Le Floch, Philippe G. LeFloch. 2023-12-29. Optimal localization for the Einstein constraints. https://arxiv.org/abs/2312.17706

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