[Paper Review] Localized initial data for Einstein equations

Yuchen Mao, Zhongkai Tao|arXiv (Cornell University)|Oct 17, 2022

Advanced Mathematical Physics Problems4 citations

TL;DR

This paper introduces a novel solution operator method to construct asymptotically flat initial data sets for the vacuum Einstein equations with improved localization and decay properties. It achieves optimal $\mathcal{O}(|x|^{2-d})$ decay and constructs nontrivial solutions supported in degenerate sectors $\{|x'| \leq x_d^\alpha\}$ for $\frac{3}{d+1} < \alpha < 1$, resolving open problems on decay rates and localization beyond cones.

ABSTRACT

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 [arXiv:1407.4766] to $\mathcal{O}(|x|^{-(d-2)})$ and a construction of nontrivial asymptotically flat initial data supported in a degenerate sector $\{(x',x_d)\in\mathbb{R}^d:|x'|\leq x_d^α\}$ for $\frac{3}{d+1}&lt;α&lt;1$.

Motivation & Objective

• To construct asymptotically flat initial data for the vacuum Einstein constraint equations with improved decay and localization.
• To resolve Carlotto's Open Problem 3.18 on achieving optimal $\mathcal{O}(|x|^{2-d})$ decay in initial data.
• To extend localization beyond conical regions to degenerate sectors $\{|x'| \leq x_d^\alpha\}$ for $\alpha \in (\frac{3}{d+1}, 1)$.
• To establish existence of smooth, nontrivial solutions supported in such anisotropic regions with precise decay estimates.
• To develop and apply a new solution operator framework using explicit fundamental solutions and weighted Sobolev spaces.

Proposed method

• The authors introduce a new solution operator $\tilde{L}$ based on averaging fundamental solutions supported on curves in degenerate sectors.
• They define anisotropic weighted Sobolev spaces $H^{s,\delta}_\alpha$ to capture the anisotropic decay behavior in the sector $\{|x'| \leq x_d^\alpha\}$.
• The method uses explicit integral representations of solution operators via curves $\gamma^{(1)}_{y,\omega}$ and $\gamma^{(2)}_{y,\omega}$ with power-law parametrization.
• A fixed-point argument is applied to the nonlinear Einstein constraint equations in the space $H^{s,\delta}_\alpha \times H^{s-1,\delta+\alpha}_\alpha$.
• The solution operators $\tilde{S}$ and $\tilde{L}$ are constructed to map into the correct weighted Sobolev spaces, ensuring well-posedness.
• The construction relies on bilinear estimates and the Banach fixed-point theorem to prove existence of a unique solution with desired regularity and decay.

Experimental results

Research questions

• RQ1 Can asymptotically flat initial data for the vacuum Einstein equations be constructed with optimal $\mathcal{O}(|x|^{2-d})$ decay?
• RQ2 Is it possible to construct nontrivial solutions supported in a degenerate sector $\{|x'| \leq x_d^\alpha\}$ for $\alpha < 1$?
• RQ3 What is the maximal extent of localization compatible with the positive mass theorem and the constraint equations?
• RQ4 Can a new solution operator framework be developed to achieve both optimal decay and improved spatial localization?
• RQ5 What weighted function spaces are necessary to capture the anisotropic decay in such degenerate sectors?

Key findings

• The paper provides a simple proof of Carlotto's Open Problem 3.18, achieving optimal $\mathcal{O}(|x|^{2-d-l})$ decay for the metric and $\mathcal{O}(|x|^{1-d-l})$ for the extrinsic curvature.
• Nontrivial asymptotically flat initial data sets are constructed that are compactly supported in a degenerate sector $\{|x'| \leq x_d^\alpha\}$ for $\frac{3}{d+1} < \alpha < 1$.
• The solution exhibits anisotropic decay: $\partial^{\beta}(g^{ij}-\delta^{ij}) = \mathcal{O}(\langle x\rangle^{1-\alpha(d-1)-|\beta'|\alpha-\beta_d})$ and $\partial^{\beta}k^{ij} = \mathcal{O}(\langle x\rangle^{1-\alpha d-|\beta'|\alpha-\beta_d})$.
• The solution space forms a smooth submanifold in a neighborhood of zero in the weighted Sobolev space $H^{s,\delta}_\alpha$.
• The method ensures $C^\infty$ regularity and optimal decay for the constructed solutions, with the decay rate matching the best possible under the positive mass theorem.
• The construction is valid for $s > \frac{d}{2}+2$, $\frac{3-(d+3)\alpha}{2} < \delta < \frac{\alpha(d-1)-3}{2}$, and $\frac{3}{d+1} < \alpha < 1$, with the range of $\alpha$ constrained by the bilinear estimate.