Localized deformation for initial data sets with the dominant energy condition

TL;DR

This paper develops a method to deform initial data sets in Einstein's equations to strictly satisfy the dominant energy condition, enabling new gluing techniques and improving understanding of energy conditions in general relativity.

Contribution

It introduces a modified constraint operator and proves local surjectivity, allowing the promotion of the dominant energy condition from weak to strict inequality.

Findings

Established local surjectivity of the modified constraint operator.

Achieved new gluing results under the dominant energy condition.

Extended previous work with refined analysis for the Einstein constraint equations.

Abstract

We consider localized deformation for initial data sets of the Einstein field equations with the dominant energy condition. Deformation results with the weak inequality need to be handled delicately. We introduce a modified constraint operator to absorb the first order change of the metric in the dominant energy condition. By establishing the local surjectivity theorem, we can promote the dominant energy condition to the strict inequality by compactly supported variations and obtain new gluing results with the dominant energy condition. The proof of local surjectivity is a modification of the earlier work for the usual constraint map by the first named author and R. Schoen and by P. Chru\'sciel and E. Delay, with some refined analysis.