Localized deformation for initial data sets with the dominant energy condition

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 [5] and by P. Chruściel and E. Delay [2], with some refined analysis.

Geometric Inequalities for Quasi-Local Masses

“…The constant γ is invariant under rescalings of the metric and hence independent of |Σ h |. Defined in (6.24), it is based on a Dirichlet energy and may be thought of as a type of capacity associated with Ω. Lastly the hypothesis of a strict dominant energy condition on horizons may be removed in many circumstances by using the localized perturbations of Corvino and Huang [17].…”

Trapped Surfaces, Topology of Black Holes, and the Positive Mass Theorem

“…More recently, we were able to provide a separate proof that avoids the use of a spin assumption [HL20b]. Our proof uses a variational argument among initial data sets satisfying the dominant energy condition, which turns out to have an intriguing connection to the question of "improving" the dominant energy condition studied by Justin Corvino and the first author [CH20]. The improvability of the dominant energy condition manifestly relates to the fundamental problem of scalar curvature deformation in differential geometry and was further explored in our recent work [HL20a].…”

Equality in the Spacetime Positive Mass Theorem

“…We now recall the modified constraint operator that was introduced by the first named author and J. Corvino in [7], based on earlier study of the modified linearization in [12, Section 6.1]. Definition 2.9.…”

“…Given a scalar function f 0 and a vector field X 0 , we introduce a functional H (see Definition 5.1) on the space of initial data sets. The functional is obtained from the classical Regge-Teitelboim Hamiltonian by replacing the usual constraint operator with the modified constraint operator Φ (g,π) introduced by the first named author and J. Corvino [7]. Choosing the pair (f 0 , X 0 ) asymptoting to (E, −2P ), we apply the Sobolev positive mass inequality (Theorem 4.1) to see that (g, k) locally minimizes the functional H among initial data sets with the dominant energy condition.…”

“…The paper is organized as follows. In Section 2, we present the basic definitions and recall the modified constraint operator of [7]. In Section 3, we present an elementary and important property of the modified constraint operator.…”