The Spacelike-Characteristic Cauchy Problem of General Relativity in Low Regularity

Abstract

In this paper we study the spacelike-characteristic Cauchy problem for the Einstein vacuum equations. Given initial data on a maximal spacelike hypersurface $\Sigma \simeq \overline{B_1} \subset {{\mathbb {R}}}^3$ and the outgoing null hypersurface ${{\mathcal {H}}}$ emanating from ${\partial }\Sigma$, we prove a priori estimates for the resulting future development in terms of low-regularity bounds on the initial data at the level of curvature in $L^2$. The proof uses the bounded $L^2$ curvature theorem [22], the extension procedure for the constraint equations [12], Cheeger-Gromov theory in low regularity [13], the canonical foliation on null hypersurfaces in low regularity [15] and global elliptic estimates for spacelike maximal hypersurfaces.

Appendices

Appendix A. Global Elliptic Estimate for the Second Fundamental Form

In this section we derive a global elliptic estimate for the second fundamental form k of a maximal spacelike hypersurface $\Sigma \simeq \overline{B_1}$. We start by deriving an integral identity for Hodge systems on compact Riemannian 3-manifolds with boundary $\Sigma$, see Lemma A.2 below. This integral identity is a slight generalisation of Section 8 in [9] where non-compact manifolds without boundary are considered.

First recall the following notation.

Definition A.1

Let $m\ge 0$ be an integer. For a given totally symmetric $(m+2)$-tensor F, define

$$\begin{aligned} A(F)_{a_1\,\dots a_{m+1}bc} := \nabla _c F_{a_1 \dots a_{m+1} b} - \nabla _b F_{a_1 \dots a_{m+1} c}, \,\, D(F)_{a_1 \dots a_{m+1}} := \nabla ^c F_{a_{1} \dots a_{m+1} c}. \end{aligned}$$

The following integral identity is a straightforward generalisation of Lemma 4.4.1 in [9] to manifolds with boundary. The proof is by integration by parts and left to the reader.

Lemma A.2

(Fundamental integral identity for Hodge systems) Let $(\Sigma ,g)$ be a compact Riemannian 3-manifold with boundary and let $m\ge 0$ be an integer. Let F be a totally symmetric $(m+2)$-tensor on $\Sigma$. Then it holds that

$$\begin{aligned} {\int \limits _\Sigma }\vert \nabla F\,\vert ^2 =&{\int \limits _\Sigma }\frac{1}{2}\vert A(F) \vert ^2 + \vert D(F) \vert ^2 \\&- {\int \limits _\Sigma }\sum \limits _{i=1}^{m+1}\Big ( \mathrm {R}_{\,\,\, a_i b}^{l\,\,\,\,\,\,\,\,\,c} F_{a_1 \dots l \dots a_{m+1}c} +\mathrm {Ric}^{l}_{\,\,\, b} F_{a_1 \dots a_{m+1}l} \Big ) F^{a_1 \dots a_{m+1} b} \\&- {\int \limits _{\partial \Sigma }}F^{a_1 \dots a_{m+1} N} D(F)_{a_1 \dots a_{m+1}} + {\int \limits _{\partial \Sigma }}\nabla _b F_{a_1 \dots a_{m+1} N} F^{a_1 \dots a_{m+1} b}. \end{aligned}$$

We now use Lemma A.2 to derive global elliptic estimates for the second fundamental form k of a maximal spacelike hypersurface. Let $({{\mathcal {M}}},\mathbf{g})$ be a vacuum spacetime and let $\Sigma \simeq \overline{B_1}$ be a compact spacelike maximal hypersurface in ${{\mathcal {M}}}$. By (2.13b), (2.13c) and (2.13d), the second fundamental form k of $\Sigma$ satisfies the following Hodge system,

$$\begin{aligned} \begin{aligned} {{\,\mathrm{div}\,}}_g k =&\,0, \\ {{{\,\mathrm{curl}\,}}}_g\,\, k=&\,H, \\ \mathrm {tr}_g \, k=&\,0. \end{aligned} \end{aligned}$$

In the notation of Definition A.1, k satisfies

$$\begin{aligned} A(k)_{iab} = \,\in ^m_{\,\,\,\, ab}H_{im}, \,\, D(k)=0. \end{aligned}$$

(A.1)

Lemma A.2 together with (A.1) yields the following corollary (see Section 8.3 in [20] for the case of manifolds without boundary).

Corollary A.3

(Fundamental global elliptic estimate for k) Let $({{\mathcal {M}}},\mathbf{g})$ be a vacuum spacetime and let $\Sigma \simeq \overline{B_1}$ be a compact spacelike maximal hypersurface in ${{\mathcal {M}}}$. Then it holds that

$$\begin{aligned} \int \limits _{\Sigma } \vert \nabla k \vert ^2 + \frac{1}{4} \vert k \vert ^4-\,\int \limits _{{\partial }\Sigma } \nabla _a k_{bN} k^{ba} \lesssim \int \limits _{\Sigma } \vert {\mathbf {R}} \vert _{{\mathbf {h}}^t}^2, \end{aligned}$$

where N denotes the outward-pointing unit normal to ${\partial }\Sigma \subset \Sigma$, T denotes the timelike unit normal to $\Sigma$ and ${\mathbf {h}}^t$ denotes the positive-definite norm on $\Sigma$ defined with T, see (2.9).

Proof

By Lemma A.2 with $F=k$, we have

$$\begin{aligned} \int \limits _{\Sigma } \vert \nabla k \vert ^2 =&\int \limits _{\Sigma } \frac{1}{2}\vert H \vert ^2 -\,\int \limits _{\Sigma } \left( \mathrm {R}^{l\,\,\,\,\,\,\,c}_{\,\,\,ab} k_{lc} + \mathrm {Ric}^l_{\,\,\, b} k_{al}\right) k^{ab} + \int \limits _{{\partial }\Sigma } \nabla _b k_{aN} k^{ab}. \end{aligned}$$

(A.2)

In dimension $n=3$, the full Riemann curvature tensor is determined by $\mathrm {Ric}$, yielding

$$\begin{aligned} \mathrm {R}^l_{\,\,\,abc} k_{lc} k^{ab} = 2 \mathrm {Ric}_{jl} k^{ji} k^{l}_{\,\,\, i} - \frac{1}{2}\mathrm {R}_{scal}\vert k \vert ^2. \end{aligned}$$

Plugging this into (A.2), we get

$$\begin{aligned} \int \limits _{\Sigma } \vert \nabla k \vert ^2 =&\int \limits _{\Sigma } \frac{1}{2}\vert H \vert ^2 -\,\int \limits _{\Sigma } \left( 3 \mathrm {Ric}^l_{\,\,\, b} k_{al} k^{ab} - \frac{1}{2}\mathrm {R}_{scal}\vert k\,\vert ^2 \right) + \int \limits _{{\partial }\Sigma } \nabla _b k_{aN} k^{ab}. \end{aligned}$$

(A.3)

On the one hand, by (2.13f) and (2.13g), we have

$$\begin{aligned} E_{ij} = \mathrm {Ric}_{ij} - k_{im}k^m_{\,\,j}, \,\, \mathrm {R}_{scal}(g) = \vert k \vert _g^2. \end{aligned}$$

On the other hand, in dimension $n=3$, it holds for symmetric tracefree 2-tensors F that

$$\begin{aligned} 3\mathrm {tr} (F^4) \ge \vert F \vert ^4. \end{aligned}$$

Hence it follows from (A.3) that

$$\begin{aligned} \int \limits _{\Sigma }\frac{1}{2}\vert H \vert ^2 =&\int \limits _{\Sigma } \vert \nabla k \vert ^2 + 3 \mathrm {Ric}_{\,\,a}^{s} k_{bs} k^{ba} - \frac{1}{2}\vert k \vert ^4 -\,\int \limits _{{\partial }\Sigma } \nabla _a k_{bN} k^{ba} \\ =&\int \limits _{\Sigma } \vert \nabla k \vert ^2 + 3 (E_{\,\,a}^{s} + k_m^{\,\, \, s} k^m_{\,\,\, a} ) k_{bs} k^{ba} - \frac{1}{2}\vert k \vert ^4 -\,\int \limits _{{\partial }\Sigma } \nabla _a k_{bN} k^{ba}\\ \ge&\int \limits _{\Sigma } \vert \nabla k \vert ^2 + 3 E_{\,\,a}^{s} k_{bs} k^{ba} + \frac{1}{2}\vert k \vert ^4 -\,\int \limits _{{\partial }\Sigma } \nabla _a k_{bN} k^{ba}. \end{aligned}$$

Using that $\vert E \vert ^2 + \vert H \vert ^2 \lesssim \vert {\mathbf {R}}\,\vert _{{\mathbf {h}}^t}^2$ by (2.8) and (2.11), we obtain

$$\begin{aligned} \int \limits _{\Sigma } \vert \nabla k \vert ^2 + \frac{1}{4} \vert k \vert ^4-\,\int \limits _{{\partial }\Sigma } \nabla _a k_{bN} k^{ba} \lesssim \int \limits _{\Sigma } \vert {\mathbf {R}} \vert _{{\mathbf {h}}^t}^2. \end{aligned}$$

This finishes the proof of Corollary A.3. $\square$

Appendix B. Proof of Lemmas 3.5 and 3.6

In this section We Prove Lemmas 3.5 and 3.6.

1.1 B.1. Proof of Lemma 3.5

We have to show that on weakly regular balls $(\Sigma ,g)$ of radius $1\le r \le 2$ with constant $0<{C_{\mathrm {ball}}}<1/2$, it holds that

$$\begin{aligned} \begin{aligned} \Vert F \Vert _{L^2({\partial }\Sigma )} \lesssim&\, \Vert F \Vert _{L^2(\Sigma )} + \Vert \nabla F \Vert _{L^2(\Sigma )}, \\ \Vert F \Vert _{L^4({\partial }\Sigma )} \lesssim&\,\Vert F \Vert _{L^2(\Sigma )} + \Vert \nabla F \Vert _{L^2(\Sigma )}, \end{aligned} \end{aligned}$$

(B.1)

and

$$\begin{aligned} \Vert F \Vert _{H^{1/2}({\partial }\Sigma )} \lesssim&\, \Vert F \Vert _{L^2(\Sigma )} + \Vert \nabla F \Vert _{L^2(\Sigma )}. \end{aligned}$$

(B.2)

First, the estimates (B.1) are straightforward, see, for example, Lemma 3.26 and Corollary 3.27 in [33] for a concise proof.

We turn to the proof of (B.2). On the one hand, by Sections 7.50 to 7.56 in [1], for each coordinate patch $U \subset {\partial }\overline{B_r}$ and smooth open set $V \subset \overline{B_r}$ with $U \subset \left( V \cap {\partial }\overline{B_r}\right)$, it holds that

$$\begin{aligned} H^1(V) \hookrightarrow H^{1/2}(U), \end{aligned}$$

where $H^{1/2}(U)$ denotes a local coordinate-defined fractional Sobolev space on U.

On the other hand, if $g_{ij} \in H^2({B_r})$ then in particular in local coordinates. By Proposition 3.2 in [30], this control suffices to compare the coordinate-defined spaces $H^{1/2}(U)$ with the space $H^{1/2}({\partial }\overline{B_r})$ defined in Definition 3.2, see also Appendix B of [30]. This finishes the proof of (B.2).This finishes the proof of Lemma 3.5.

1.2 B.2. Proof of Lemma 3.6

Let $1\le r_0 \le 2$ and $0<{C_{\mathrm {ball}}}<1/2$ be two real numbers. Let $(\Sigma ,g)$ be a weakly regular ball of radius $r_0$ with constant ${C_{\mathrm {ball}}}$. Let F be a tensor on $\Sigma$. We have to show that

$$\begin{aligned} \Vert F \Vert _{L^2(\Sigma )} \lesssim \Vert \nabla F \Vert _{L^2(\Sigma )} + \Vert F \Vert _{L^2({\partial }\Sigma )}. \end{aligned}$$

Define spherical coordinates $(r,\theta ^1, \theta ^2)$ with $r \in [0,r_0]$ and $(\theta ^1,\theta ^2)\in {{\mathbb {S}}}^2$ on $\Sigma \simeq \overline{B_{r_0}}$ as in Section 2.4. Let ${\gamma }$ and $d \mu _{\gamma }$ on ${{\mathcal {S}}}_r$ denote the standard round metric of radius $r>0$ and its volume element, respectively. Denote by ${\overset{\circ }{\gamma }}$ and $d \mu _{{\overset{\circ }{\gamma }}}$ the standard round metric on the unit sphere and its volume element, respectively. Note that $\gamma = r^2 {\overset{\circ }{\gamma }}$. By the fundamental theorem of calculus and using that ${\partial }\Sigma = {{\mathcal {S}}}_{r_0}$, we have for $0<r \le r_0$,

$$\begin{aligned} \int \limits _{{{\mathcal {S}}}_r} \vert F \vert _g^2 \, d\mu _{\overset{\circ }{\gamma }}=&\int \limits _{r_0}^{r} {\partial }_{r'} \left( \, \int \limits _{{{\mathcal {S}}}_{r'}} \vert F \vert ^2_g d\mu _{\overset{\circ }{\gamma }}\right) dr' + \int \limits _{{\partial }\Sigma } \vert F \vert _g^2 d\mu _{\overset{\circ }{\gamma }}\\ =&\int \limits _{r_0}^r \left( \, \int \limits _{{{\mathcal {S}}}_{r'}} \nabla _{{\partial }_{r'}} F \cdot F \, d\mu _{\overset{\circ }{\gamma }}\right) dr' + \int \limits _{{\partial }\Sigma } \vert F \vert _g^2 d\mu _{\overset{\circ }{\gamma }}\\ \le&\left\| \vert {\partial }_r \vert _g^2 \right\| _{L^\infty (\Sigma )} \left( \int \limits _{r_0}^r \int \limits _{{{\mathcal {S}}}_{r'}} \vert \nabla F \vert ^2_g d\mu _{\overset{\circ }{\gamma }}dr' \right) ^{1/2} \left( \int \limits _{r_0}^r \int \limits _{{{\mathcal {S}}}_{r'}} \vert F \vert ^2_g d\mu _{\overset{\circ }{\gamma }}dr' \right) ^{1/2}\\&+ \int \limits _{{\partial }\Sigma } \vert F \vert _g^2 d\mu _{\overset{\circ }{\gamma }}. \end{aligned}$$

Using that by definition of the spherical coordinates on $B_{r_0}$ and the weakly regular ball property of $(\Sigma ,g)$,

$$\begin{aligned} \vert {\partial }_r \vert _g^2 = g_{rr} = \frac{x^i}{r} \frac{x^j}{r} g_{ij} \le (1+ {C_{\mathrm {ball}}})\frac{x^i}{r} \frac{x^j}{r} e_{ij} =1+{C_{\mathrm {ball}}}, \end{aligned}$$

we can estimate the right-hand side as follows,

$$\begin{aligned} \int \limits _{{{\mathcal {S}}}_r} \vert F \vert _g^2 \, d\mu _{\overset{\circ }{\gamma }}\lesssim&\left( \int \limits _{r_0}^r \int \limits _{{{\mathcal {S}}}_{r'}} \vert \nabla F \vert ^2_g d\mu _{\overset{\circ }{\gamma }}dr' \right) ^{1/2} \left( \int \limits _{r_0}^r \int \limits _{{{\mathcal {S}}}_{r'}} \vert F \vert ^2_g d\mu _{\overset{\circ }{\gamma }}dr' \right) ^{1/2}+ \int \limits _{{\partial }\Sigma } \vert F \vert _g^2 d\mu _{\overset{\circ }{\gamma }}\\ \lesssim&\,\frac{1}{r^2} \left( \int \limits _{r_0}^r \int \limits _{{{\mathcal {S}}}_{r'}} \vert \nabla F \vert ^2_g d\mu _{\gamma } dr' \right) ^{1/2} \left( \int \limits _{r_0}^r \int \limits _{{{\mathcal {S}}}_{r'}} \vert F \vert ^2_g d\mu _{\gamma } dr' \right) ^{1/2}+ \int \limits _{{\partial }\Sigma } \vert F \vert _g^2 d\mu _{\overset{\circ }{\gamma }}\\ \lesssim&\, \frac{1}{r^2} \left( \, \int \limits _{B_{r_0}} \vert \nabla F \vert ^2_g d\mu _{e} \right) ^{1/2} \left( \, \int \limits _{B_{r_0}} \vert F \vert ^2_g d\mu _{e} \right) ^{1/2} + \int \limits _{{\partial }\Sigma } \vert F \vert _g^2 d\mu _{\overset{\circ }{\gamma }}\\ \lesssim&\,\frac{1}{r^2} \Vert \nabla F \Vert _{L^2(B_{r_0})}\Vert F \Vert _{L^2(B_{r_0})} + \int \limits _{{\partial }\Sigma } \vert F \vert _g^2 d\mu _{\overset{\circ }{\gamma }}, \end{aligned}$$

where $d\mu _e$ denotes the measure with respect to the standard Euclidean metric e in Cartesian coordinates, and we used in the last inequality that $\Sigma$ is a weakly regular ball to compare the Euclidean integral with the g-dependent norm.

Multiplying the above by $r^2$ and using that by Lemmas 3.1, 3.5 and 3.7,

$$\begin{aligned} \int \limits _{{\partial }\Sigma } \vert F \vert _g^2 d\mu _{r_0^2{\overset{\circ }{\gamma }}} \lesssim \Vert F \Vert _{L^2({\partial }\Sigma )}^2, \end{aligned}$$

we get that for $0<r\le r_0$,

$$\begin{aligned} \begin{aligned} \int \limits _{{{\mathcal {S}}}_r} \vert F \vert _g^2 \, d\mu _\gamma \lesssim&\,\Vert \nabla F \Vert _{L^2(\Sigma )} \Vert F \Vert _{L^2(\Sigma )} + \frac{r^2}{r_0^2} \int \limits _{{\partial }\Sigma } \vert F \vert _g^2 d\mu _{r_0^2{\overset{\circ }{\gamma }}} \\ \lesssim&\, \Vert \nabla F \Vert _{L^2(\Sigma )} \Vert F \Vert _{L^2(\Sigma )} + \frac{r^2}{r_0^2} \Vert F\Vert ^2_{L^2({\partial }\Sigma )}. \end{aligned} \end{aligned}$$

(B.3)

By (B.3) and using that $(\Sigma ,g)$ is a weakly regular ball of radius $r_0$, we get that

$$\begin{aligned} \Vert F \Vert _{L^2(\Sigma )}^2&\lesssim \int \limits _{B_{r_0}} \vert F \vert ^2_g d\mu _e \lesssim \int \limits _{0}^{r_0} \left( \, \int \limits _{{{\mathcal {S}}}_r} \vert F \vert _g^2 \, d\mu _\gamma \right) dr \\&\lesssim \Vert \nabla F \Vert _{L^2(\Sigma )} \Vert F \Vert _{L^2(\Sigma )} + \Vert F\Vert ^2_{L^2({\partial }\Sigma )}, \end{aligned}$$

which implies that

$$\begin{aligned} \Vert F \Vert _{L^2(\Sigma )} \lesssim \Vert \nabla F \Vert _{L^2(\Sigma )} + \Vert F \Vert _{L^2({\partial }\Sigma )}. \end{aligned}$$

This finishes the proof of Lemma 3.6.

Appendix C. Proof of Lemma 5.4

In this section we prove Lemma 5.4. Let (M, g) be a smooth compact Riemannian 3-manifold with boundary such that $M \simeq \overline{B_1} \subset {{\mathbb {R}}}^3$ and for a real number $1\le t \le 2$,

$$\begin{aligned} \mathrm {Ric}= 0 \text { on } M, \,\, \mathrm {tr}\, \Theta =\frac{2}{t}, \, {\widehat{\Theta }}=0 \text { on } {\partial }M. \end{aligned}$$

(C.1)

We have to prove that

$$\begin{aligned} (M,g) \cong (\overline{B_t},e). \end{aligned}$$

First, by the Gauss equation

$$\begin{aligned} 2K = \left( \mathrm {tr}\Theta \right) ^2- \vert \Theta \vert ^2 + R_{\mathrm {scal}} - 2 \mathrm {Ric}(N,N), \end{aligned}$$

it follows that the Gauss curvature of is constant $K=\frac{1}{t^2}$. Hence by classical differential geometry, there exist coordinates $(\theta ^1, {\theta }^2)$ on ${\partial }M$ such that

(C.2)

Second, by the smoothness of (M, g), there exists in an open neighbourhood ${{\mathcal {U}}}\subset M$ of ${\partial }M \subset M$ a so-called Gaussian coordinate system $({r},{\theta }^1,{\theta }^2)$, see for example Section 3.3 in [36], which coincide with (C.2) on ${\partial }M$ and are such that for some small real number $\delta >0$,

$$\begin{aligned}&{{\mathcal {U}}}= \{ {r}\in (t-\delta ,t]\}, \\&{{\mathcal {U}}}\cap {\partial }M = \{ {r}=t\}, \\&\nabla _{{\partial }_r}{\partial }_r =0, \\&{\partial }_r \vert _{{\partial }M} \text { is normal to } {\partial }M, \\&g({\partial }_r,{\partial }_r)=1. \end{aligned}$$

In such coordinates we can express g as

where denotes the induced metric on the level sets $S_r$ of r. In particular, by (C.2) it holds that

Third, we claim that the Riemannian manifold (M, g) smoothly extends onto ${{\mathbb {R}}}^3 \setminus \overline{B_t}$ when identifying ${\partial }M = {\partial }\overline{B_t}\subset {{\mathbb {R}}}^3$. It suffices to show that the induced metric

is smooth across $\{r=t\}$. Here $\gamma _{AB}(r,\theta ^1,\theta ^2)$ is the standard round metric of radius r. In the following calculations we denote on ${{\mathcal {U}}}$ simply by .

By (C.2) it follows that is continuous across $\{r=t\}$. Further, on the one hand, on ${\partial }\overline{B_t}\subset {{\mathbb {R}}}^3$, it holds that

$$\begin{aligned} {\partial }_r \gamma _{AB}\vert _{r=t} =&2 t {\overset{\circ }{\gamma }}_{AB}, \,\, {\partial }_r^2 \gamma _{AB}\vert _{r=t} = 2 {\overset{\circ }{\gamma }}_{AB}, \,\, {\partial }_r^{m} \gamma _{AB}\vert _{r=t}=0 \text { for } m\ge 3, \end{aligned}$$

(C.3)

where ${\overset{\circ }{\gamma }}_{AB}$ denotes the metric components of the standard round metric on ${\mathbb {S}}^2$.

On the other hand, by construction of $(r,\theta ^1, \theta ^2)$ it holds on ${{\mathcal {U}}}$ that

(C.4)

By (C.1), (C.2) and (C.4), on ${\partial }M$ we thus have that

(C.5)

Differentiating (C.4) in r yields

(C.6)

where we used that $\mathrm {Rm}=0$ in (M, g) and $\nabla _{{\partial }_r}{\partial }_r=0$ in Gaussian coordinates in ${{\mathcal {U}}}$. From (C.5) and (C.6) and using that $a=1$, it follows that

(C.7)

Differentiating (C.4) further in r shows that

(C.8)

Comparing (C.3) with (C.5), (C.7) and (C.8) shows that all r-derivatives of agree on $r=t$. Hence (M, g) smoothly extends as Riemannian manifold onto $({{\mathbb {R}}}^3 \setminus \overline{B_t},e)$.

The resulting smooth Riemannian 3-manifold is in particular flat, complete and has cubic volume growth of geodesic balls. By Proposition 4.4 in [13] it must therefore be isometric to $({{\mathbb {R}}}^3,e)$. We conclude that by the above construction,

$$\begin{aligned} (M,g) \cong ({{\mathbb {R}}}^3 \setminus \left( {{\mathbb {R}}}^3 \setminus \overline{B_t}\right) ,e) \cong (\overline{B_t},e), \end{aligned}$$

which finishes the proof of Lemma 5.4.

Appendix D. Comparison Estimates Between Maximal Spacelike Foliations

In this section we prove Lemma 4.6. By assumption, for real numbers $D>0$ and $\varepsilon >0$, ${{\mathcal {M}}}_{t^*}$ is foliated by maximal spacelike hypersurface $(\Sigma _t)_{1\le t\,\le t^*}$ given as level sets of a time function t with $\Sigma _1 = \Sigma$ and satisfying

$$\begin{aligned} \begin{aligned}&\Vert \mathbf{R}\Vert _{L^\infty _{t} L^2({\Sigma }_{t})} + \Vert \mathrm {Ric}\Vert _{L^\infty _tL^2(\Sigma _t)} \le \,D\varepsilon , \\&\Vert \nabla k \Vert _{L^\infty _{ t} L^2({\Sigma }_{ t})} + \Vert {k} \Vert _{L^\infty _{ t} L^2({\Sigma }_{ t})} \le \, D\varepsilon , \\&\Vert n-1 \Vert _{L^\infty ({{\mathcal {M}}}_{t^*})} + \Vert \nabla n \Vert _{L^\infty _tL^2(\Sigma _t)} +\Vert \nabla ^2 n \Vert _{L^\infty _tL^2(\Sigma _t)}\lesssim \,D\varepsilon . \end{aligned} \end{aligned}$$

(D.1)

Let T denote the timelike unit normal to $\Sigma _t$.

Moreover, by assumption there is another foliation on ${{\mathcal {M}}}_{t^*}$ (constructed by the bounded $L^2$ curvature theorem) of maximal hypersurfaces $(\widetilde{\Sigma }_{{{\tilde{t}}}})_{0 \le {{\tilde{t}}}\,\le t^*}$ given as level sets of a time function ${{\tilde{t}}}$ with $\Sigma _{t^*} = \widetilde{\Sigma }_{t^*}$ and satisfying

$$\begin{aligned} \begin{aligned}&\Vert \mathbf{R}\Vert _{L^\infty _{{{\tilde{t}}}} L^2(\widetilde{\Sigma }_{{{\tilde{t}}}})} + \Vert \widetilde{\mathrm {Ric}}\Vert _{L^\infty _{{{\tilde{t}}}} L^2(\widetilde{\Sigma }_{{{\tilde{t}}}})} \lesssim \,{C_{\mathrm {ball}}}, \\&\Vert {{\widetilde{\nabla }}}{\tilde{k}} \Vert _{L^\infty _{{{\tilde{t}}}} L^2(\widetilde{\Sigma }_{{{\tilde{t}}}})} + \Vert {\tilde{k}} \Vert _{L^\infty _{{{\tilde{t}}}} L^2(\widetilde{\Sigma }_{{{\tilde{t}}}})}+ \Vert \mathbf{D}\widetilde{\pi } \Vert _{L^\infty _{{\tilde{t}}}L^2({{\widetilde{\Sigma }}}_{{\tilde{t}}})} \lesssim \,{C_{\mathrm {ball}}}, \\&\Vert {\tilde{n}}-1 \Vert _{L^\infty ({{\mathcal {M}}}_{t^*})} + \Vert {{\widetilde{\nabla }}}{\tilde{n}} \Vert _{L^\infty ({{\mathcal {M}}}_{t^*})} + \Vert {{\widetilde{\nabla }}}^2 {\tilde{n}} \Vert _{L^\infty _{{\tilde{t}}}L^2(\widetilde{\Sigma }_{{\tilde{t}}})} \lesssim \, {C_{\mathrm {ball}}}, \end{aligned} \end{aligned}$$

(D.2)

where $\widetilde{\mathrm {Ric}}$, ${{\widetilde{\nabla }}}$ and ${\tilde{k}}$ denote the Ricci curvature, covariant derivative and the second fundamental form on $\widetilde{\Sigma }_{{{\tilde{t}}}}$, respectively. Let ${{\widetilde{T}}}$ denote the timelike unit normal to $\widetilde{\Sigma }_{{{\tilde{t}}}}$.

We need to show that for $\varepsilon >0$ and ${C_{\mathrm {ball}}}>0$ sufficiently small, for $1\le t \le t^*$,

$$\begin{aligned} \left\| {{\tilde{\nu }}}-1\right\| _{L^\infty _t L^\infty (\Sigma _t)} \lesssim&\,{C_{\mathrm {ball}}}, \end{aligned}$$

(D.3)

$$\begin{aligned} \Vert \widetilde{k} \Vert _{L^\infty _t L^4(\Sigma _t)}\,\lesssim&\, {C_{\mathrm {ball}}}, \end{aligned}$$

(D.4)

where the angle ${\tilde{\nu }}$ between T and ${{\widetilde{T}}}$ is defined as

$$\begin{aligned} {{\tilde{\nu }}}:= -\mathbf{g}(T, {{\widetilde{T}}}). \end{aligned}$$

(D.5)

The proof of (D.3) and (D.4) is based on a standard continuity argument going backwards in t and starting at $\Sigma _{t^*}={{\widetilde{\Sigma }}}_{t^*}$ where by construction

$$\begin{aligned} {{\tilde{\nu }}}=1, \,\, k={{\tilde{k}}}. \end{aligned}$$

In the following, we only discuss the bootstrap assumption and its improvement.

Bootstrap assumption. Let $1\le t_0^*<t^*$ be a real. Assume that for a large constant $M>0$, for $t_0^*\le t \le t^*$,

$$\begin{aligned} \begin{aligned} \left\| {{\tilde{\nu }}}-1\right\| _{L^\infty _tL^\infty (\Sigma _t)} \le M {C_{\mathrm {ball}}}. \end{aligned}\end{aligned}$$

(D.6)

First consequences of the bootstrap assumption. Let $({{\tilde{e}}}_i)_{i=1,2,3}$ be an orthonormal frame on $\widetilde{\Sigma }_{{\tilde{t}}}$ and let ${{\tilde{e}}}_0= {{\widetilde{T}}}$. Expressing T as

$$\begin{aligned} T={{\tilde{\nu }}}{{\widetilde{T}}}+ \mathbf{g}(T, {\tilde{e}}_i) {\tilde{e}}_i, \end{aligned}$$

and using that $\mathbf{g}(T,T)=-1$, it follows that

$$\begin{aligned} -1 = - \vert {{\tilde{\nu }}}\vert ^2 + \sum \limits _{i=1,2,3} \vert \mathbf{g}(T,{{\tilde{e}}}_i)\vert ^2. \end{aligned}$$

This implies by the bootstrap assumption (D.6) for ${C_{\mathrm {ball}}}>0$ sufficiently small that for $t_0^*\le t \le t^*$,

$$\begin{aligned} \, \left\| \mathbf{g}(T,{{\tilde{e}}}_i)\right\| _{L^\infty _tL^\infty (\Sigma _t)} \lesssim \sqrt{M{C_{\mathrm {ball}}}}. \end{aligned}$$

(D.7)

Let $(e_i)_{i=1,2,3}$ be the orthonormal frame tangent to $\Sigma _t$ constructed by the Gram-Schmidt method applied to the $\Sigma _t$-tangential frame

$$\begin{aligned} ({{\tilde{e}}}_1+\mathbf{g}({{\tilde{e}}}_1,T)T,{{\tilde{e}}}_2+\mathbf{g}({{\tilde{e}}}_2,T)T,{{\tilde{e}}}_3+\mathbf{g}({{\tilde{e}}}_3,T)T), \end{aligned}$$

and set moreover $e_0 := T$. By the Gram-Schmidt construction and (D.6) and (D.7), we get that for ${C_{\mathrm {ball}}}>0$ sufficiently small, for $t_0^*\le t \le t^*$,

$$\begin{aligned} \begin{aligned}&\sum \limits _{i=1}^3 \left( \left\| \mathbf{g}(e_i,{{\tilde{e}}}_i)-1\right\| _{L^\infty _tL^\infty (\Sigma _t)} +\sum \limits _{j\ne i} \left\| \mathbf{g}({{\tilde{e}}}_i,e_j)\right\| _{L^\infty _tL^\infty (\Sigma _t)} + \left\| \mathbf{g}({{\widetilde{T}}},e_i)\right\| _{L^\infty _tL^\infty (\Sigma _t)} \right) \\&\quad \lesssim \sqrt{M{C_{\mathrm {ball}}}}. \end{aligned} \end{aligned}$$

(D.8)

Improvement of the bootstrap assumption. First, by definition of ${{\tilde{\nu }}}$, see (D.5), and using that $\mathbf{D}_{T}T = n^{-1}\nabla n$,

$$\begin{aligned} \begin{aligned} T ({{\tilde{\nu }}}) =&-\mathbf{g}(\mathbf{D}_{T} T,{{\widetilde{T}}}) -\mathbf{g}(T,\mathbf{D}_{T}{{\widetilde{T}}}) \\ =&- \left( \mathbf{g}({{\widetilde{T}}},e_i)\mathbf{g}(\mathbf{D}_{T}T,e_i) + \mathbf{g}(T,{{\tilde{e}}}_i) \mathbf{g}({{\tilde{e}}}_i, \mathbf{D}_{\mathbf{g}(T,{{\tilde{e}}}_j){{\tilde{e}}}_j-\mathbf{g}(T,{{\tilde{e}}}_0){{\tilde{e}}}_0} {{\widetilde{T}}}) \right) \\ =&-\left( \mathbf{g}({{\widetilde{T}}},e_i)(n^{-1}\nabla _{e_i} n) - \mathbf{g}(T,{{\tilde{e}}}_j) \mathbf{g}(T,{{\tilde{e}}}_i) \widetilde{k}_{ij} + {{\tilde{\nu }}}\mathbf{g}(T,{{\tilde{e}}}_i) (\widetilde{n}^{-1}\widetilde{\nabla }_{{{\tilde{e}}}_i}\widetilde{n}) \right) . \end{aligned} \end{aligned}$$

(D.9)

Define shift-free coordinates $(t,x^1,x^2,x^3)$ on ${{\mathcal {M}}}_{t^*}$ by transporting $(x^1,x^2,x^3)$ from $\Sigma _{t^*}$ backwards along T. Then it holds that ${\partial }_t = n^{-1} T$, and we get from (D.9) that

$$\begin{aligned} {\partial }_t {{\tilde{\nu }}}= -n^{-1}\left( \mathbf{g}({{\widetilde{T}}},e_i)(n^{-1}\nabla _{e_i} n) - \mathbf{g}(T,{{\tilde{e}}}_j) \mathbf{g}(T,{{\tilde{e}}}_i) \widetilde{k}_{ij} + {{\tilde{\nu }}}\mathbf{g}(T,{{\tilde{e}}}_i) (\widetilde{n}^{-1}\widetilde{\nabla }_{{{\tilde{e}}}_i}\widetilde{n}) \right) . \end{aligned}$$

Integrating this equation in t, using that ${{\tilde{\nu }}}\vert _{t=t^*}=1$ and applying (D.1), (D.2), (D.6), (D.7), (D.8) and Lemma 3.4, we get that for $\varepsilon >0$ and ${C_{\mathrm {ball}}}>0$ sufficiently small, for $t^*_0 \le t \le t^*$, it holds that

$$\begin{aligned} \begin{aligned} \left\| {{\tilde{\nu }}}-1\right\| _{L^\infty _t L^4(\Sigma _t)} \lesssim&\, \Vert {\mathbf{g}({{\widetilde{T}}},e_i)}\Vert _{L^\infty ({{\mathcal {M}}}_{[t_0^*,t^*]})}\Vert {\nabla _{e_i} n}\Vert _{L^4({{\mathcal {M}}}_{[t_0^*,t^*]})} \\&+ \Vert {\mathbf{g}(T,{{\tilde{e}}}_i)}\Vert _{L^\infty ({{\mathcal {M}}}_{[t_0^*,t^*]})} \Vert {\mathbf{g}(T,{{\tilde{e}}}_j)}\Vert _{L^\infty ({{\mathcal {M}}}_{[t_0^*,t^*]})}\Vert {\widetilde{k}_{ij}}\Vert _{L^4({{\mathcal {M}}}_{[t_0^*,t^*]})} \\&+ \Vert {{{\tilde{\nu }}}}\Vert _{L^\infty ({{\mathcal {M}}}_{[t_0^*,t^*]})}\Vert {\mathbf{g}(T,{{\tilde{e}}}_i)}\Vert _{L^\infty ({{\mathcal {M}}}_{[t_0^*,t^*]})}\Vert {\widetilde{\nabla }_{{{\tilde{e}}}_i}\widetilde{n}}\Vert _{L^4({{\mathcal {M}}}_{[t_0^*,t^*]})} \\ \lesssim&\, \Vert {\mathbf{g}({{\widetilde{T}}},e_i)}\Vert _{L^\infty ({{\mathcal {M}}}_{[t_0^*,t^*]})}\\&\left( \Vert {\nabla n}\Vert _{L^\infty _tL^2(\Sigma _t)}+\Vert {\nabla ^2 n}\Vert _{L^\infty _tL^2(\Sigma _t)} \right) \\&+ \Vert {\mathbf{g}(T,{{\tilde{e}}}_i)}\Vert _{L^\infty ({{\mathcal {M}}}_{[t_0^*,t^*]})} \Vert {\mathbf{g}(T,{{\tilde{e}}}_j)}\Vert _{L^\infty ({{\mathcal {M}}}_{[t_0^*,t^*]})}\\&\left( \Vert {\widetilde{k}}\Vert _{L^\infty _{{\tilde{t}}}L^2(\widetilde{\Sigma }_{{\tilde{t}}})}+\Vert {\widetilde{\nabla } \widetilde{k}}\Vert _{L^\infty _{{\tilde{t}}}L^2(\widetilde{\Sigma }_{{\tilde{t}}})} \right) \\&+ \Vert {{{\tilde{\nu }}}}\Vert _{L^\infty ({{\mathcal {M}}}_{[t_0^*,t^*]})}\Vert {\mathbf{g}(T,{{\tilde{e}}}_i)}\Vert _{L^\infty ({{\mathcal {M}}}_{[t_0^*,t^*]})}\\&\left( \Vert {\widetilde{\nabla }\widetilde{n}}\Vert _{L^\infty _{{\tilde{t}}}L^2(\widetilde{\Sigma }_{{\tilde{t}}})}+\Vert {\widetilde{\nabla }^2 \widetilde{n}}\Vert _{L^\infty _{{\tilde{t}}}L^2(\widetilde{\Sigma }_{{\tilde{t}}})} \right) \\ \lesssim&\, \sqrt{M{C_{\mathrm {ball}}}} (D\varepsilon ) + \sqrt{M{C_{\mathrm {ball}}}} {C_{\mathrm {ball}}}, \end{aligned} \end{aligned}$$

(D.10)

where ${{\mathcal {M}}}_{[t_0^*,t^*]}$ denotes the spacetime region $t_0^*\le t \le t^*$.

Second, by definition of ${{\tilde{\nu }}}$ in (D.5), for $i=1,2,3$,

$$\begin{aligned} \nabla _{e_i}{{\tilde{\nu }}}=&-\mathbf{g}(\mathbf{D}_{e_i}T,{{\widetilde{T}}}) - \mathbf{g}(T,\mathbf{D}_{e_i}{{\widetilde{T}}}) \\ =&- \left( \mathbf{g}({{\widetilde{T}}},e_j)k_{ij} + \mathbf{g}(T,{{\tilde{e}}}_j) \mathbf{g}({{\tilde{e}}}_j, \mathbf{D}_{\mathbf{g}(e_i,{{\tilde{e}}}_l){{\tilde{e}}}_l-\mathbf{g}(e_i,{{\tilde{e}}}_0){{\tilde{e}}}_0} {{\widetilde{T}}}) \right) \\ =&- \left( \mathbf{g}({{\widetilde{T}}},e_j)k_{ij} - \mathbf{g}(T,{{\tilde{e}}}_j) \mathbf{g}(e_i,{{\tilde{e}}}_l) \widetilde{k}_{lj}+ \mathbf{g}(T,{{\tilde{e}}}_j) \mathbf{g}(e_i,{{\widetilde{T}}}) (\widetilde{n}^{-1} \widetilde{\nabla }_{{{\tilde{e}}}_j} \widetilde{n}) \right) \end{aligned}$$

By (D.1), (D.2), (D.8) and Lemma 3.4, we have for ${C_{\mathrm {ball}}}>0$ and $\varepsilon >0$ sufficiently small,

$$\begin{aligned} \begin{aligned} \left\| \nabla _{e_i} {{\tilde{\nu }}}\right\| _{L^\infty _t L^4(\Sigma _t)} \lesssim&\, \Vert {\mathbf{g}({{\widetilde{T}}},e_j)}\Vert _{L^\infty ({{\mathcal {M}}}_{[t_0^*,t^*]})}\Vert {k_{ij}}\Vert _{L^\infty _t L^4(\Sigma _t)} \\&+ \Vert {\mathbf{g}(T,{{\tilde{e}}}_j)}\Vert _{L^\infty ({{\mathcal {M}}}_{[t_0^*,t^*]})}\Vert {\mathbf{g}(e_i,{{\tilde{e}}}_l)}\Vert _{L^\infty ({{\mathcal {M}}}_{[t_0^*,t^*]})}\Vert {\widetilde{k}_{lj}}\Vert _{L^\infty _t L^4(\Sigma _t)} \\&+ \Vert {\mathbf{g}(T,{{\tilde{e}}}_j)}\Vert _{L^\infty ({{\mathcal {M}}}_{[t_0^*,t^*]})}\Vert {\mathbf{g}(e_i,{{\widetilde{T}}})}\Vert _{L^\infty ({{\mathcal {M}}}_{[t_0^*,t^*]})}\Vert {\widetilde{\nabla }_{{{\tilde{e}}}_j} \widetilde{n}}\Vert _{L^\infty _t L^\infty (\Sigma _t)} \\ \lesssim&\sqrt{M{C_{\mathrm {ball}}}} D\varepsilon + \sqrt{M{C_{\mathrm {ball}}}}\Vert { \widetilde{k}_{lj}}\Vert _{L^\infty _t L^4(\Sigma _t)} +(M{C_{\mathrm {ball}}}) {C_{\mathrm {ball}}}. \end{aligned} \end{aligned}$$

(D.11)

By (D.10), (D.11) and Lemma 3.4, it follows that

$$\begin{aligned} \left\| {{\tilde{\nu }}}-1\right\| _{L^\infty ({{\mathcal {M}}}_{[t_0^*,t^*]})} \lesssim \sqrt{M{C_{\mathrm {ball}}}} (D\varepsilon +\sqrt{M{C_{\mathrm {ball}}}}{C_{\mathrm {ball}}}) + \sqrt{M{C_{\mathrm {ball}}}} \Vert {\widetilde{k}_{lj} }\Vert _{L^\infty _t L^4(\Sigma _t)}. \end{aligned}$$

(D.12)

To estimate the remaining term $\Vert {\widetilde{k}_{lj} }\Vert _{L^\infty _t L^4(\Sigma _t)}$ on the right-hand side of (D.12), we apply the following technical lemma whose proof is postponed to the end of this section.

Lemma 5.5

(Technical lemma) Under the assumption of (D.1), (D.2), (D.6) for ${C_{\mathrm {ball}}}>0$ and $\varepsilon >0$ sufficiently small, it holds for every scalar function f on ${{\mathcal {M}}}_{t^*}$ that for $t_0^*\le t \le t^*$,

$$\begin{aligned} \left\| f\right\| _{L^\infty _t L^4(\Sigma _t)} \lesssim&\left\| \mathbf{D}f \right\| _{L^\infty _{{\tilde{t}}}L^2({{\widetilde{\Sigma }}}_{{\tilde{t}}})}^{1/4} \left\| f\right\| _{L^\infty _{{\tilde{t}}}L^6({{\widetilde{\Sigma }}}_{{\tilde{t}}})}^{3/4} + \left\| f\right\| _{L^4(\Sigma _{t^*})}. \end{aligned}$$

Applying Lemma 5.5 to $f = \widetilde{k}_{ij}$, we have for $\varepsilon >0$ and ${C_{\mathrm {ball}}}>0$ sufficiently small, for $t_0^*\le t \le t^*$,

$$\begin{aligned} \begin{aligned} \Vert \widetilde{k}_{lj}\Vert _{L^\infty _t L^4(\Sigma _t)} \lesssim&\, \Vert \mathbf{D}(\widetilde{k}_{lj})\Vert _{L^\infty _{{\tilde{t}}}L^2({{\widetilde{\Sigma }}}_{{\tilde{t}}})}^{1/4} \Vert \widetilde{k}_{lj}\Vert ^{3/4}_{L^\infty _{{\tilde{t}}}L^6({{\widetilde{\Sigma }}}_{{\tilde{t}}})} +\Vert \widetilde{k}_{lj}\Vert _{L^4(\Sigma _{t^*})}\\ \lesssim&\left( \Vert \mathbf{D}\widetilde{\pi } \Vert _{L^\infty _{{\tilde{t}}}L^2({{\widetilde{\Sigma }}}_{{\tilde{t}}})} + \Vert \widetilde{\pi } \Vert _{L^\infty _{{\tilde{t}}}L^4({{\widetilde{\Sigma }}}_{{\tilde{t}}})}\Vert \widetilde{{\mathbf {A}}} \Vert _{L^\infty _{{\tilde{t}}}L^4({{\widetilde{\Sigma }}}_{{\tilde{t}}})} \right) ^{1/4} \Vert \widetilde{k}\Vert ^{3/4}_{L^\infty _{{\tilde{t}}}L^6({{\widetilde{\Sigma }}}_{{\tilde{t}}})} \\&+\Vert k\Vert _{L^4(\Sigma _{t^*})}\\ \lesssim&\, {C_{\mathrm {ball}}}^{1/4} {C_{\mathrm {ball}}}^{3/4} + D\varepsilon , \end{aligned} \end{aligned}$$

(D.13)

where $\widetilde{{\mathbf {A}}}$ denotes the connection 1-form defined by

$$\begin{aligned} (\widetilde{{\mathbf {A}}}_\mu )_{{\alpha }{\beta }} := \mathbf{g}(\mathbf{D}_{{\tilde{e}}_\mu } {\tilde{e}}_{\beta }, {\tilde{e}}_{\alpha }), \,\, \text { for }\mu ,{\alpha },{\beta }=0,1,2,3, \end{aligned}$$

and we used that the foliation $(\widetilde{\Sigma }_{{{\tilde{t}}}})_{0 \le {{\tilde{t}}}\,\le t^*}$ constructed by the bounded $L^2$ curvature theorem (Theorem 3.11, see also [22]) satisfies in addition to (D.2) the following bound on $0\le {{\tilde{t}}}\le t^*$,

$$\begin{aligned} \Vert \widetilde{{\mathbf {A}}} \Vert _{L^\infty _{{\tilde{t}}}L^4({{\widetilde{\Sigma }}}_{{\tilde{t}}})} \lesssim {C_{\mathrm {ball}}}. \end{aligned}$$

Plugging (D.13) into (D.12), we get that for ${C_{\mathrm {ball}}}>0$ and $\varepsilon >0$ sufficiently small, for $t_0^*\le t \le t^{*}$,

$$\begin{aligned} \left\| {{\tilde{\nu }}}-1\right\| _{L^\infty _tL^\infty (\Sigma _t)} \lesssim&\, \sqrt{M{C_{\mathrm {ball}}}}(D\varepsilon +{C_{\mathrm {ball}}}+\sqrt{M{C_{\mathrm {ball}}}}{C_{\mathrm {ball}}}) \\ \le&\, M'{C_{\mathrm {ball}}}, \end{aligned}$$

for a constant $0<M'<M$. This improves the bootstrap assumption (D.6) and hence finishes the proof of (D.3). The estimate (D.4) follows directly from (D.13). This finishes the proof of Lemma 4.6.

It remains to prove Lemma 5.5. Let $(t,x^1,x^2,x^3)$ denote the shift-free coordinate system constructed above. Using that $\mathrm {tr}k =0$ on $\Sigma _t$ for $1\le t \le t^*$, (D.1), (D.2) and ${\partial }_t = n^{-1} T$, we have for ${C_{\mathrm {ball}}}>0$ and $\varepsilon >0$ sufficiently small, on $t^*_0 \le t \le t^*$,

$$\begin{aligned} \int \limits _{\Sigma _{t}} f^4 \,d\mu _{g} =&\int \limits _{t^*}^{t} {\partial }_{t'}\left( \, \int \limits _{\Sigma _{t'}}f^4 \,d\mu _g \right) dt' + \int \limits _{\Sigma _{t^*}} f^4 \,d\mu _{g} \\ =&\, 4\int \limits _{t^*}^{t } \left( \, \int \limits _{\Sigma _{t'}}{\partial }_t f f^3 \,d\mu _{g} \right) dt' + \int \limits _{\Sigma _{t^*}} f^4 \,d\mu _{g}\\ \lesssim&\int \limits _{{{\mathcal {M}}}_{[t_0^*,t^*]}} \vert \mathbf{D}f\vert _{{\mathbf {h}}^t} \vert f \vert ^3 + \int \limits _{\Sigma _{t^*}} f^4 \,d\mu _{g} \\ \lesssim&\int \limits _{{{\mathcal {M}}}_{[t_0^*,t^*]}} \vert \mathbf{D}f\vert _{{\mathbf {h}}^{{\tilde{t}}}} \vert f \vert ^3 + \int \limits _{\Sigma _{t^*}} f^4 \,d\mu _{g} \\ \lesssim&\left\| \mathbf{D}f\right\| _{L^\infty _{{\tilde{t}}}L^2({{\widetilde{\Sigma }}}_{{\tilde{t}}})} \left\| f\right\| _{L^\infty _{{\tilde{t}}}L^6({{\widetilde{\Sigma }}}_{{\tilde{t}}})}^3+ \int \limits _{\Sigma _{t^*}} f^4 \,d\mu _{g}, \end{aligned}$$

where we used (D.6) to compare the positive-definite norms ${\mathbf {h}}^{{\tilde{t}}}$ and ${\mathbf {h}}^t$. This finishes the proof of Lemma 5.5.