Abstract
We prove a harmonic asymptotics density theorem for asymptotically flat initial data sets with compact boundary that satisfy the dominant energy condition. We use this to settle the spacetime positive mass theorem, with rigidity, for initial data sets with apparent horizon boundary in dimensions less than 8 without a spin assumption.
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Notes
Note that this discussion only appears in the arXiv version of the paper.
More specifically, this Kerr initial data comes from an element of the “reference family” for Kerr, as described in [CD03].
A bounded linear operator \(T:X\rightarrow Y\) is semi-Fredholm if \(\dim \ker T<\infty \) and T(X) is closed in Y.
In this theorem and throughout the paper, whenever we refer to Hölder spaces on M, we mean that they are regular up to the boundary.
In the case when \({\text {tr}}_g k\) has a good sign, see [Met10, Theorem 3.4].
The graphical components tend to \(\pm \infty \) on approach to these cylinders. We say that the Jang graph “blows up" over the MOTS or MITS.
The \(n=3\) claim follows from observing that the only term appearing in \({\text {tr}}_{g_j}k_j\) (when written in terms of g, k, \(\lambda \), and the deformations u, Yh, w) that is not directly controlled is the \({\text {tr}}_g k\) term, which is controlled by assumption. See [Eic13, Proposition 15].
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Acknowledgements
We thank Lan-Hsuan Huang for useful discussions at the start of this project, Greg Galloway for his interest in the problem, and Piotr Chruściel for various helpful comments.
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Appendix A. Second Differential of the Constraint-Null Expansion System
Appendix A. Second Differential of the Constraint-Null Expansion System
In this paper, we utilize the inverse function theorem to perturb families of initial data sets. To this end, we need to control the constants appearing in the “quantitative" version of the inverse function theorem [Lee19, Theorem A.43].
Lemma A.1
Let \((M^n,g, \pi )\) be an asymptotically flat data set as in Sect. 2.1. Let \(K_1\subset T_{(1,0)}{\mathcal {C}}\) be a closed subspace and \(K_2\subset T_{(g,\pi )}{\mathcal {D}}\) be a finite-dimensional subspace. There exists a constant \(C_0\) such that for any \(r_0>0\) sufficiently small, the following is true.
Let \((\gamma ,\tau )\in {\mathcal {D}}\) with \(\Vert (\gamma ,\tau )-(g,\pi )\Vert _{\mathcal {D}}\le r_0\) and define
Then
and
for any \(((u,Y),(h,w))\in B_{r_0}((1,0),(0,0))\).
This lemma also holds if in the definition of \(\hat{{\mathcal {P}}}\), we use the modified constraint operator \(\overline{\Phi }_{(g,\pi )}\) instead of \(\Phi \).
Here \(L_2(X,Y)\) refers to the space of bounded multilinear maps \(X\times X\rightarrow Y\). Note that a Lipschitz bound for \(D\hat{{\mathcal {P}}}_{(\gamma ,\tau )}\) follows from the Hessian bound by the mean value theorem in Banach spaces. The proof proceeds with a computation of \(D\Phi ,D^2\Phi ,D\Theta \), and \(D^2\Theta \).
Lemma A.2
The first derivative (linearization) of the constraint operator is given by
Schematically, the second derivative is given by
Here we use the usual schematic notation where \(A*B\) denotes linear combinations and contractions of the components of A and B with respect to the metric g.
The schematic notation misses factors of g and \(g^{-1}\) but these are pointwise bounded by Morrey’s inequality. In the following calculation, we use the shorthand \(\delta _g F=DF|_{g}(h)\).
Proof
The formula for \(D\Phi \) is well known in the literature [FM73]. It depends on the linearization of the scalar curvature, which can be found in [Lee19], for instance. To obtain the formula for \(D^2\Phi \), we simply differentiate (A.3), making note of the following rules:
-
\(\delta _g \nabla T=\nabla h_2*T + \nabla \delta _g T\) for any tensor T, and
-
contractions produce terms of \(h_2\) \(*\) what was being contracted.
Finally, we also note that the variation of the Ricci tensor is given by
where \(\Delta _L\) is the Lichnerowicz Laplacian. In our schematic notation, this becomes
The variation in \(\pi \) is much more straightforward and (A.4) is easily obtained along these lines. \(\square \)
Lemma A.3
The first derivative of the boundary null expansion is given by
where \(\omega _i=h_{ij}\nu ^j-h(\nu ,\nu )\nu _i\) and \(\nu ^\flat \) denotes the 1-form dual to \(\nu \). Schematically, the second derivative is given by
where schematic notation here is omitting terms like \(\nu \) and H.
Proof
We first compute the linearization of the normal. Varying \(g(\nu ,\nu )=1\) gives
while varying \(g(X,\nu )=0\) for \(X\in T\Sigma \) gives
It follows that
Secondly, we compute the linearization of the second fundamental form. For X and Y tangent to \(\Sigma \), we have
Taking the variation, we have
Now
so that finally
The mean curvature of the boundary is given by
so taking the variation and using (A.8) yields
The formula for \(D\Theta \) follows easily, where also note that
The schematic computation for \(D^2\Theta \) also follows easily using the rules establised in the proof of Lemma A.2. \(\square \)
From these formulas, we deduce:
Lemma A.4
There exists a constant \(C_0\) such that for any sufficiently small \(r_0>0\) the following is true. If \((\gamma ,\tau )\in {\mathcal {D}}\) satisfies \(\Vert (\gamma ,\tau )-(g,\pi )\Vert _{\mathcal {D}}\le r_0\), then
and
Proof
We first remark that the constants appearing in the Sobolev, Morrey, and trace inequalities associated to the metric \(\gamma \) can be bounded in terms of \(r_0\). The first estimate (A.10) can be read off from the explicit formulas (A.3) and (A.5). For example, consider
We rewrite this as
and from this it is not hard to see that the \(L^p_{-q}\) norm can be estimated by \(\lesssim r_0 \Vert h\Vert _{W^{2,p}_{-q}}\).
To prove the estimate (A.11), we examine the bilinear structure of the schematic formulas (A.4) and (A.6). For \(D^2\Phi \), we put the highest number of derivatives in \(L^p_{-q}\) and the lowest number of derivatives in \(L^\infty \) using Morrey’s inequality. Special care must be taken with the \(\mathrm {Riem}*h_1*h_2\) term, as the curvature is not assumed to be pointwise bounded. However, it is in \(L^p_{-q}\), so we just put \(h_1\) and \(h_2\) in \(L^\infty \). Altogether, we obtain the estimate
For \(D^2\Theta \), we estimate each of the terms appearing in (A.6) in \(W^{1-\frac{1}{p},p}(\partial \Omega )\). Terms with derivatives are handled using Lemma 2.4 instead of Morrey’s inequality. Note that our schematic notation omits the normal \(\nu _g\) and mean curvature \(H_g\), however both of these are pointwise bounded in terms of \(\gamma \). Therefore, we obtain the estimate
as desired. \(\square \)
We can now prove the main result of this appendix, Lemma A.1.
Proof of Lemma A.1
We first define a function
so that
By the chain rule for functions on Banach spaces,
The second derivative is given by
The derivatives of \(\hat{\Psi }_{(\gamma ,\tau )}\) are given by
and
In these formulas, the differentials are being evaluated at ((u, Y), (h, w)) or \({\hat{\Psi }}_{(\gamma ,\tau )}((u,Y),(h,w))\), wherever appropriate.
To prove (A.1), we use (A.12) for \((\gamma ,\tau )\) and \((g,\pi )\) at ((1, 0), (0, 0)), which yields
For \((\gamma ,\tau )\) sufficiently close to \((g,\pi )\), we may evidently estimate both of these terms (in operator norm) using (A.14) and the estimate (A.10).
To prove (A.2), we note that (A.13) implies
For ((u, Y), (h, w)) small, \(\hat{\Psi }_{(\gamma ,\tau )}((u,Y),(h,w))\) is close to \((g,\pi )\) in \({\mathcal {D}}\), so we may apply (A.10) and (A.11). Furthermore, the same estimates may be derived for \(D{\hat{\Psi }}\) and \(D^2{\hat{\Psi }}\) from (A.14) and (A.15). This completes the proof of (A.2).\(\square \)
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Lee, D.A., Lesourd, M. & Unger, R. Density and Positive Mass Theorems for Initial Data Sets with Boundary. Commun. Math. Phys. 395, 643–677 (2022). https://doi.org/10.1007/s00220-022-04439-1
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DOI: https://doi.org/10.1007/s00220-022-04439-1