Localized initial data for Einstein equations

. We apply a new method with explicit solution operators to construct asymptotically flat initial data sets of the vacuum Einstein equation with new localization properties. Applications include an improvement of the decay rate in Carlotto–Schoen [CS16] to O ( | x | − ( d − 2) ). Also we construct nontrivial asymptotically flat initial data supported in a degenerate sector { ( x ′ , x d ) ∈ R d : | x ′ | ≤ x αd } for 3 d +1 < α < 1.

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