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Density and Positive Mass Theorems for Initial Data Sets with Boundary

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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

  1. Note that this discussion only appears in the arXiv version of the paper.

  2. More specifically, this Kerr initial data comes from an element of the “reference family” for Kerr, as described in [CD03].

  3. A bounded linear operator \(T:X\rightarrow Y\) is semi-Fredholm if \(\dim \ker T<\infty \) and T(X) is closed in Y.

  4. 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.

  5. In the case when \({\text {tr}}_g k\) has a good sign, see [Met10, Theorem 3.4].

  6. 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.

  7. 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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Authors and Affiliations

  1. Graduate Center and Queens College, City University of New York, 365 Fifth Avenue, New York, NY, 10016, USA

    Dan A. Lee

  2. Black Hole Initiative, Harvard University, Cambridge, MA, 02138, USA

    Martin Lesourd

  3. Department of Mathematics, Princeton University, Princeton, NJ, 08544, USA

    Ryan Unger

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  1. Dan A. Lee
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Correspondence to Martin Lesourd.

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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

$$\begin{aligned}\hat{{\mathcal {P}}}_{(\gamma ,\tau )}:[(1,0)+K_1]\times K_2&\rightarrow {\mathcal {L}}\times W^{1-\frac{1}{p},p}(\partial M)\\ ((u,Y),(h,w))&\mapsto (\Phi ,\Theta )[\Psi _{(\gamma ,\tau )}(u,Y)+(h,w)].\end{aligned}$$

Then

$$\begin{aligned} \Vert D\hat{{\mathcal {P}}}_{(\gamma ,\tau )}|_{((1,0),(0,0))}-D\hat{{\mathcal {P}}}_{(g,\pi )}|_{((1,0),(0,0))}\Vert _{L(K_1\times K_2,{\mathcal {L}}\times W^{1-\frac{1}{p},p})}\le C_0r_0\end{aligned}$$
(A.1)

and

$$\begin{aligned} \Vert D^2 \hat{{\mathcal {P}}}_{(\gamma ,\tau )}|_{((u,Y),(h,w))}\Vert _{L_2(K_1\times K_2,{\mathcal {L}}\times W^{1-\frac{1}{p},p})}\le C_0 \end{aligned}$$
(A.2)

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

$$\begin{aligned} D\Phi |_{(g,\pi )}(h,w)&=\bigg (-\Delta _g({\text {tr}}_g h)+{\text {div}}_g({\text {div}}_g h)-\langle \mathrm {Ric}_g,h\rangle _g+\tfrac{2}{n-1}({\text {tr}}_g\pi )\left( \pi ^{ij}h_{ij}+{\text {tr}}_g w\right) \nonumber \\&\qquad - 2g_{kl}\pi ^{ik}\pi ^{jl}h_{ij}-2\langle \pi ,w\rangle _g,\nonumber \\&\qquad ({\text {div}}_g w)^i-\tfrac{1}{2} g^{ij}\pi ^{kl}\nabla _j h_{kl}+g^{ij}\pi ^{kl}\nabla _ kh_{jl}+\tfrac{1}{2} \pi ^{ij}\nabla _j ({\text {tr}}_g h)\bigg ) \end{aligned}$$
(A.3)

Schematically, the second derivative is given by

$$\begin{aligned} D^2\Phi |_{(g,\pi )}((h_1,w_1),(h_2,w_2))&=\bigg (\sum _{0\le i_1+i_2\le 2}\nabla ^{i_1}h_1*\nabla ^{i_2}h_2+\mathrm {Riem}* h_1 * h_2\nonumber \\&\qquad +\pi *\pi *h_1*h_2+w_1*w_2+\pi *h_1*w_2+\pi *h_2*w_1,\nonumber \\&\qquad w_1*\nabla h_2+w_2*\nabla h_1+ \pi * h_1*\nabla h_2+\pi * \nabla h_1*h_2\bigg ). \end{aligned}$$
(A.4)

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

$$\begin{aligned}-2\delta _g R_{ij}=\Delta _Lh_{2\,ij}+\nabla _i\nabla _j {\text {tr}}_g h_2-\nabla _i({\text {div}}_g h_2)_j-\nabla _j({\text {div}}_g h_2)_i,\end{aligned}$$

where \(\Delta _L\) is the Lichnerowicz Laplacian. In our schematic notation, this becomes

$$\begin{aligned}\delta _g \mathrm {Ric}=\nabla ^2h_2+\mathrm {Riem}* h_2.\end{aligned}$$

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

$$\begin{aligned}&D\Theta |_{(g,\pi )}(h,w)=\tfrac{1}{2} {\text {tr}}_{\partial M}(\nabla _\nu h) -{\text {div}}_{\partial M} \omega - \tfrac{1}{2} h(\nu ,\nu )H\nonumber \\&\quad -h(\nu ,\nu )\pi (\nu ^\flat ,\nu ^\flat )-w(\nu ^\flat ,\nu ^\flat ), \end{aligned}$$
(A.5)

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

$$\begin{aligned}&D^2\Theta |_{(g,\pi )}((h_1,w_1),(h_2,w_2))=\sum _{0\le i_1+i_2\le 1}\nabla ^{i_1}h_1*\nabla ^{i_2}h_2\nonumber \\&\quad +h_1*w_2+h_2*w_1+w_1*w_2, \end{aligned}$$
(A.6)

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

$$\begin{aligned}h(\nu ,\nu )+2g(\nu ,\delta _g\nu )=0,\end{aligned}$$

while varying \(g(X,\nu )=0\) for \(X\in T\Sigma \) gives

$$\begin{aligned}h(X,\nu )+g(X,\delta _g\nu )=0.\end{aligned}$$

It follows that

$$\begin{aligned} \delta _g \nu ^i=-h^{ij}\nu _j+\tfrac{1}{2} h(\nu ,\nu )\nu ^i. \end{aligned}$$
(A.7)

Secondly, we compute the linearization of the second fundamental form. For X and Y tangent to \(\Sigma \), we have

$$\begin{aligned}A(X,Y)=-g(\nu ,\nabla _X Y)=-g_{ij} \nu ^i X^k \nabla _k Y^j.\end{aligned}$$

Taking the variation, we have

$$\begin{aligned} \delta _g A(X,Y)&=-h_{ij}\nu ^i X^k\nabla _k Y^j-g_{ij}\left( -h^{il}\nu _l+\tfrac{1}{2} h(\nu ,\nu )\nu ^i\right) X^k\nabla _k Y^j-g_{ij}\nu ^i X^k\delta _g(\nabla _kY^j)\\&= \tfrac{1}{2} h(\nu ,\nu )\left( -g_{ij}\nu ^i X^k\nabla _k Y^j\right) -g_{ij}\nu ^i X^k\delta _g(\nabla _kY^j)\\&=\tfrac{1}{2} h(\nu ,\nu )A(X,Y)-\tfrac{1}{2} g_{ij}\nu ^i g^{jm} \left( \nabla _k h_{lm} +\nabla _l h_{km}-\nabla _m h_{kl}\right) X^k Y^l\\&=\tfrac{1}{2}\left( h(\nu ,\nu )A_{kl}-\nu ^m\nabla _k h_{lm}-\nu ^m\nabla _l h_{km}+\nu ^m\nabla _m h_{kl}\right) X^kY^l. \end{aligned}$$

Now

$$\begin{aligned}\nu ^m\nabla _k h_{lm}=\nabla _k \omega _l +h(\nu ,\nu )A_{kl} - h_{ln}A_k{}^n,\end{aligned}$$

so that finally

$$\begin{aligned} \delta _g A(X,Y)= \tfrac{1}{2}(\nabla _\nu h_{ij}-\nabla ^{\partial M}_i\omega _j -\nabla _j^{\partial M}\omega _i+h_{ik}A_j{}^k +h_{jk}A_i{}^k - h(\nu ,\nu )A_{ij})X^iY^j.\nonumber \\ \end{aligned}$$
(A.8)

The mean curvature of the boundary is given by

$$\begin{aligned}H={\text {tr}}_{\partial M}A=(g^{ij}-\nu ^i\nu ^j)A_{ij},\end{aligned}$$

so taking the variation and using (A.8) yields

$$\begin{aligned} \delta _g H=\tfrac{1}{2} {\text {tr}}_{\partial M}(\nabla _\nu h) -{\text {div}}_{\partial M} \omega - \tfrac{1}{2} h(\nu ,\nu )H.\end{aligned}$$
(A.9)

The formula for \(D\Theta \) follows easily, where also note that

$$\begin{aligned}\delta _g\nu ^\flat = \tfrac{1}{2} h(\nu ,\nu )\nu ^\flat .\end{aligned}$$

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

$$\begin{aligned} \Vert D(\Phi ,\Theta )|_{(\gamma ,\tau )}-D(\Phi ,\Theta )|_{(g,\pi )}\Vert \le C_0r_0 \end{aligned}$$
(A.10)

and

$$\begin{aligned} \Vert D^2(\Phi ,\Theta )|_{(g,\pi )}\Vert \le C_0. \end{aligned}$$
(A.11)

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

$$\begin{aligned}g^{ij}\partial _i\partial _j (g^{kl}h_{kl})-\gamma ^{ij}\partial _i\partial _j(\gamma ^{kl}h_{kl}).\end{aligned}$$

We rewrite this as

$$\begin{aligned}(g^{ij}-\gamma ^{ij})\partial _i\partial _j(\gamma ^{kl}h_{kl})+g^{ij}\partial _i\partial _j((g^{kl}-\gamma ^{kl})h_{kl})\end{aligned}$$

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

$$\begin{aligned}\Vert D^2\Phi |_{(\gamma ,\tau )}((h_1,w_1),(h_2,w_2))\Vert _{\mathcal {L}}\lesssim \Vert (h_1,w_1)\Vert _{W^{2,p}_{-q}\times W^{1,p}_{-q-1}}\Vert (h_2,w_2)\Vert _{W^{2,p}_{-q}\times W^{1,p}_{-q-1}}.\end{aligned}$$

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

$$\begin{aligned}\Vert D^2\Theta |_{(\gamma ,\tau )}\Vert _{W^{1-\frac{1}{p},p}}\lesssim \Vert (h_1,w_1)\Vert _{W^{2,p}_{-q}\times W^{1,p}_{-q-1}}\Vert (h_2,w_2)\Vert _{W^{2,p}_{-q}\times W^{1,p}_{-q-1}}, \end{aligned}$$

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

$$\begin{aligned} {\hat{\Psi }}_{(\gamma ,\tau )}:[(1,0)+K_1]\times K_2&\rightarrow {\mathcal {D}}\\ ((u,Y),(h,w))&\mapsto \Psi _{(\gamma ,\tau )}(u,Y)+(h,w), \end{aligned}$$

so that

$$\begin{aligned}\hat{{\mathcal {P}}}_{(\gamma ,\tau )}=(\Phi ,\Theta )\circ \hat{\Psi }_{(\gamma ,\tau )}.\end{aligned}$$

By the chain rule for functions on Banach spaces,

$$\begin{aligned} D\hat{{\mathcal {P}}}_{(\gamma ,\tau )}((v_1,Z_1),(h_1,w_1))=D(\Phi ,\Theta )\circ D{\hat{\Psi }}_{(\gamma ,\tau )}((v_1,Z_1),(h_1,w_1)).\end{aligned}$$
(A.12)

The second derivative is given by

$$\begin{aligned}&D^2\hat{{\mathcal {P}}}_{(\gamma ,\tau )}\Big (((v_1,Z_1),(h_1,w_1)),((v_2,Z_2),(h_2,w_2))\Big )\nonumber \\&\quad =D^2(\Theta ,\Phi )\Big (D{\hat{\Psi }}_{(\gamma ,\tau )}((v_1,Z_1),(h_1,w_1)),D{\hat{\Psi }}_{(\gamma ,\tau )}((v_2,Z_2),(h_2,w_2))\Big )\nonumber \\&\qquad +D(\Theta ,\Phi )\circ D^2{\hat{\Psi }}_{(\gamma ,\tau )}\Big (((v_1,Z_1),(h_1,w_1)),((v_2,Z_2),(h_2,w_2))\Big ). \end{aligned}$$
(A.13)

The derivatives of \(\hat{\Psi }_{(\gamma ,\tau )}\) are given by

$$\begin{aligned} D\hat{\Psi }_{(\gamma ,\tau )}((v_1,Z_1),(h_1,w_1))= & {} (su^{s-1}v_1\gamma +h_1, -\tfrac{3}{2} s u^{-\frac{3}{2} s-1}v(\tau +{\mathfrak {L}}_\gamma Y)\nonumber \\&+ u^{-\frac{3}{2}s}{\mathfrak {L}}_\gamma Z_1+w_1) \end{aligned}$$
(A.14)

and

$$\begin{aligned}&D^2\hat{\Psi }_{(\gamma ,\tau )}\Big (((v_1,Z_1),(h_1,w_1)),((v_2,Z_2),(h_2,w_2))\Big )=(s(s-1)u^{s-2}v_1v_2,\nonumber \\&\quad \tfrac{3}{2} s(\tfrac{3}{2} s+1)u^{-\frac{3}{2} s-2}v_1v_2(\tau +{\mathfrak {L}}_\gamma Y)-\tfrac{3}{2} s u^{-\frac{3}{2} s-1}v_2{\mathfrak {L}}_\gamma Z_1 -\tfrac{3}{2} s u^{-\frac{3}{2} s-1}v_1{\mathfrak {L}}_\gamma Z_2 ).\nonumber \\ \end{aligned}$$
(A.15)

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

$$\begin{aligned} D\hat{{\mathcal {P}}}_{(\gamma ,\tau )}-D\hat{{\mathcal {P}}}_{(g,\pi )}= & {} D(\Phi ,\Theta )|_{(\gamma ,\tau )}(D{\hat{\Psi }}_{(\gamma ,\tau )}-D{\hat{\Psi }}_{(g,\pi )})+(D(\Phi ,\Theta )|_{(\gamma ,\tau )}\\&-D(\Phi ,\Theta )|_{(g,\pi )})D{\hat{\Psi }}_{(g,\pi )}. \end{aligned}$$

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

$$\begin{aligned}\Vert D^2\hat{{\mathcal {P}}}_{(\gamma ,\tau )}|_{((u,Y),(h,w))}\Vert&\le \left\| D^2(\Theta ,\Phi )|_{{\hat{\Psi }}_{(\gamma ,\tau )}((u,Y),(h,w))}\right\| \cdot \left\| D{\hat{\Psi }}_{(\gamma ,\tau )}|_{((u,Y),(h,w))}\right\| ^2\\&\quad \quad + \left\| D(\Theta ,\Phi )|_{{\hat{\Psi }}_{(\gamma ,\tau )}((u,Y),(h,w))}\right\| \cdot \left\| D^2{\hat{\Psi }}_{(\gamma ,\tau )}|_{((u,Y),(h,w))}\right\| .\end{aligned}$$

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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