[Paper Review] Initial data gluing in the asymptotically flat regime via solution operators with prescribed support properties

Yuchen Mao, Sung‐Jin Oh+1 more|arXiv (Cornell University)|Aug 24, 2023

Stochastic processes and financial applications4 citations

TL;DR

This paper presents a new, simplified proof of initial data gluing for asymptotically flat Riemannian manifolds using explicit solution operators with prescribed support properties for the linearized constraint equations around flat space. The approach enables optimal regularity, improved decay, and positivity conditions, yielding strengthened versions of known gluing theorems—including obstruction-free gluing—via purely spacelike techniques, bypassing prior reliance on null gluing methods.

ABSTRACT

We give new proofs of general relativistic initial data gluing results on unit-scale annuli based on explicit solution operators for the linearized constraint equation around the flat case with prescribed support properties. These results retrieve and optimize - in terms of positivity, regularity, size and/or spatial decay requirements - a number of known theorems concerning asymptotically flat initial data, including Kerr exterior gluing by Corvino-Schoen and Chruściel-Delay, interior gluing (or "fill-in") by Bieri-Chruściel, and obstruction-free gluing by Czimek-Rodnianski. In particular, our proof of the strengthened obstruction-free gluing theorem relies on purely spacelike techniques, rather than null gluing as in the original approach.

Motivation & Objective

• To provide a new, streamlined proof of initial data gluing on unit-scale annuli in the asymptotically flat regime.
• To optimize regularity, decay, positivity, and size constraints in known gluing theorems for asymptotically flat initial data.
• To establish a strengthened obstruction-free gluing theorem using purely spacelike techniques, avoiding reliance on null gluing methods.
• To construct localized initial data sets with prescribed charges and support via conic-type solution operators and nonlinear positivity arguments.

Proposed method

• Construction of explicit Bogovskii-type solution operators for the linearized constraint equations on annular domains with zero boundary conditions.
• Use of conic-type solution operators to generate initial data localized in conic regions, inspired by Carlotto–Schoen and Aretakis–Czimek–Rodnianski.
• Application of nonlinear computations akin to Bartnik’s mass positivity argument in the time-symmetric, almost-flat case.
• Employment of a localized boost argument based on the fundamental theorem of Choquet-Bruhat and Chruściel’s computation.
• Use of rescaling to extend unit-scale gluing results to the asymptotically flat regime.
• Derivation of decay estimates for perturbations of the Kerr metric and their derivatives in terms of mass, angular momentum, and boost parameters.

Experimental results

Research questions

• RQ1 Can initial data gluing in the asymptotically flat regime be proven using only spacelike techniques, without relying on null gluing?
• RQ2 What is the optimal level of Sobolev regularity achievable in initial data gluing theorems with prescribed support?
• RQ3 How can solution operators for the linearized constraint equations be constructed with exact support properties to enable localized gluing?
• RQ4 What are the minimal decay and positivity requirements for obstruction-free gluing of asymptotically flat initial data?
• RQ5 Can the construction of initial data with prescribed charges be made explicit and localized using solution operators?

Key findings

• A new, short proof of initial data gluing on unit-scale annuli is achieved using explicit solution operators with prescribed support, retrieving and optimizing known results.
• The method achieves optimal (modulo endpoint) Sobolev regularity for the gluing construction, improving upon prior approaches.
• The obstruction-free gluing theorem is strengthened via purely spacelike techniques, eliminating dependence on null gluing and optimizing positivity, regularity, and decay.
• Localized initial data sets with prescribed charges are constructed using conic-type solution operators and nonlinear positivity arguments, enabling precise control over support and mass.
• Decay estimates for perturbations of the Kerr metric and their derivatives are derived, showing bounds of order $|y|^{-1}$ for mass and boost derivatives, and $|y|^{-2}$ for angular momentum derivatives.
• The approach generalizes to other divergence-type equations with variable coefficients, as indicated by the authors’ upcoming work [24].