Proof. Under hypothesis (b), Theorem 3.7 and Corollary 5.2 of [8] together yield the statement for a neighbourhood of \(\{\gamma(r)\}_{r>0}\), assuming only that \(\gamma\) is a ray. However, in the case of a line the Busemann function \(b^+\) is independent of which point on \(\gamma\) is selected to be \(\gamma(0)\), apart from an additive constant. In case (b) this establishes (i); while (ii)–(iii) follow from Lemma 3.3–3.4 (ibid), completeness of \(\gamma\), and compactness of \(X\), using a covering argument. As Galloway and Horta note, (iii) and (i) are even easier to prove in case (a), as was first done by Eschenburg [14] and exploited in [15]; (ii) holds as soon as \(X \subset I^-(\gamma(R)) \cap I^+(\gamma(-R))\) in case (a), and the existence of such \(R\) follows from compactness of \(X \subset U\) by a covering argument since finiteness of \(b^\pm\) implies \(U\subset I(\gamma)\). ◻

Proof. The definition 5 of \(b^+_r\), maximality of \(\sigma_r:[0,s_r] \longrightarrow M\) between \(\sigma_r(0)\) and \(\sigma_r(s_r)=\gamma(r)\), and \(s \in (0,s_r]\) yield \[\begin{align} u_r(x) &= \ell(\gamma(0),\sigma_r(s_r)) - \ell(\sigma_r(0),\sigma_r(s_r)) +\ell(\sigma_r(0),\sigma_r(s))- \ell(x,\sigma_r(s)) \nonumber \\ &= \ell(\gamma(0),\sigma_r(s_r)) - \ell(\sigma_r(s),\sigma_r(s_r)) - \ell(x,\sigma_r(s)) \nonumber \\ & \ge \ell(\gamma(0),\sigma_r(s_r))- \ell(x,\sigma_r(s_r)) \nonumber \\ & =b^+_r(x) \nonumber \end{align}\] as desired, where the reverse triangle inequality 4 has been used.

For \(a \in [0,s]\), equality holds at \(x=\sigma_r(a)\) holds since maximality of \(\sigma_r\) ensures the triangle inequality is saturated. The fact that \(\gamma\) maximizes time from \(S\) to \(\gamma(r)\) implies \(0<\ell(x,\sigma_r(s_r)) \le r\) for each \(x \in I^+(S)\cap I^-(\sigma_r(s_r))\). The reverse triangle inequality shows \(\ell(x,\sigma_r(s))\), \(u_r(x)\) and \(b^+_r(x)\) are all finite on the smaller neighbourhood \(I^+(S)\cap I^-(\sigma_r(s))\) of \(\sigma_r(0)\). ◻

Proof. For \(R\) large enough (depending on \(\gamma(0)\)) we shall show \((b^+_r)_{r \ge R}\) has a semiconcavity constant \(C\) of which depends on \(\gamma\) but not on \(\gamma(0)\). Fix a compact set \(X \subset U\). Theorem 2(ii) provides \(R\) and \(\tilde{C}\) such that for each \(x \in X\) and \(z =\gamma(r)\) with \(r \ge R\), there is a (proper-time parameterized) maximimizing geodesic segment \(\sigma_r\) joining \(\sigma_r(0)=x\) to \(\sigma_r(a_r)=z\). Theorem 2(iii) asserts \(|\dot{\sigma}_r(0)|_{\tilde{g}} \le \tilde{C}\) for all \(x \in X\) and \(r \ge R\). Since \(a_r=\ell(x,\gamma(r)) \ge \ell(x,\gamma(R)) + r-R\) diverges as \(r \to \infty\), taking \(R\) larger if necessary ensures \(a_r \ge 1\) (uniformly with respect to \((x,r) \in X \times [R,\infty)\)).

Now \(K_{\tilde{C}} := \{ (v,y) \in TM \mid |v|_g=1, |v|_{\tilde{g}} \le \tilde{C}, y \in X\}\) is a compact set of unit timelike directions. Among timelike geodesics with initial conditions in \(K_{\tilde{C}}\), the time to the timelike cut-locus attains its minimum over \(K_{\tilde{C}}\) by the lower semicontinuity shown in [2] for case (a). This mininum value is positive; call it \(t_0 \in (0,\infty]\) and fix \(0<t<\min\{t_0,1\}\). The set \(G = \{\exp_z sv \mid (v,z,s) \in K_{\tilde{C}} \times [0,t]\}\) is then compact and contains \(\sigma_r({[0, t]})\) for all \(x \in X\) and \(r \ge R\).

Given \(\sigma_r\) as above, fix \(y=\sigma_r(t)\). Lemma 1 asserts \(u(x') = b^+_{r}(x)+\ell(x,y)-\ell(x',y) \ge b^+_{r}(x')\) holds for all \(x'\) in a neighbourhood of \(x\) and equality holds at \(x'=x\); in other words, \(u\) supports \(b^+_{r}\) from above at \(x\). Thus \(b^+_{r}\) inherits from \(u\) at \(x\) a semiconcavity constant \(C(x,y)\) which depends continuously on its arguments in \(\{\ell>0\}\) for case (a), according to Proposition 3.4 of [9]. Taking \(C = \max_{(x,y) \in G} C(x,y)\) then concludes the proof of case (a).

Although the proposition last mentioned was proved under hypothesis (a), the conclusions we need can be extended to hypothesis (b) by the following observation. The proof of Proposition 3.4 of [9] provides a semiconcavity constant \(C\) of \(b^+_r\) at \(x=\sigma_r(0)\) given by a simple integral which depends only on the geometry \((\tilde{g},g)\) of \(M\) along the proper-time parameterized timelike geodesic \(\{\sigma_r(s)\}_{s \in [0,t]}\). Since this geodesic lies in the compact set \(G\), this explicit formula is easy to bound, allowing us to conclude the same way whenever \(t_0>0\). If \(t_0=0\), we set \(t=1\) and argue as before after observing the resulting set \(G\) is again compact since the exponential map is continuous and defined globally on the timelike tangent bundle in case (b). ◻

Proof. Semiconcavity implies that in any smooth coordinate chart, the function becomes a concave function after subtraction of a multiple \(C\) of a fixed parabola, depending on the coordinates. Equi-semiconcavity of \(\{b^+_r\}_{r \ge R}\) therefore implies locally uniform convergence of \(b^+_r \to b^+\) on \(U\) and convergence a.e.of their gradients, as for concave functions; c.f.Theorem 10.9 and the proof of Theorem 25.7 from Rockafellar [47]. The limit \(b^+\) inherits the same constant \(C\) of semiconcavity. Semiconcavity also implies differentiability of \(b^+_r\) outside a set of measure zero. For approximate Busemann functions 5 , in the globally hyperbolic case (a) it is well-known that \(|db^\pm_r|=1\) where defined; Theorem 3.6 of [9]. In case (b) this also holds since we already know that \(b_r^+\) is a.e.differentiable on \(U\) and for any \(q \in U\) there exists a timelike maximizer to \(\gamma(r)\) (for large enough \(r\), uniformly on compact subsets of \(U\), by Theorem 2(ii)). Thus \(|db^+|=1\) a.e.on \(U\). The same estimates apply to \(-b^-\). The reverse triangle inequality implies \[\label{1-steep} b^\pm_r(y) - b^\pm_r(x) \ge \ell(x,y)\tag{11}\] for all \(y \in J^+(x)\), so \(db^\pm_r\) and their limits are both future-directed (recall that the Lorentzian metric has signature \((+,-,\dots,-)\)). ◻

Proof. We write \(u := \ell(\cdot,o)\). By partition of unity, we may and will assume \(\phi \in C^1_0(U)\) is supported on a fixed chart. Since \(u\) is semiconvex, partitioning further if necessary, we may also assume the existence of a smooth function \(v\) such that \(u+v\) is a convex function of the coordinates in the usual Euclidean sense [48] . It is also not restrictive to assume that the coordinate representation \((2-p)\,({\partial}^i u)\,({\partial}^j u)/\vert du\vert^2 - g^{ij}\) of the tensor from 10 is uniformly positive-definite on the support of \(\phi\). By [48], the constructions below will not depend on the choice of \(v\). To relax our notation, we regard \(M\) as being covered by a single chart from which it inherits a Euclidean metric.

Standard distribution theory implies the distributional Euclidean Hessian of \(u+v\) is given by a nonnegative-definite matrix-valued Radon measure \(D^2(u+v)\). In turn, as \(v\) is smooth the distributional Hessian of \(u\) is a signed Radon measure \(D^2u\) satisfying \[\begin{align} \int_M \partial_i\partial_j\phi\,u\, d x = \int_M \phi\,d(D^2u)_{ij}. \end{align}\] Moreover, the \(dx\)-singular part \((D^2 u)^{\perp}\) of \(D^2u\) is nonnegative-definite.

Given \(\varepsilon > 0\) let \(\rho_\varepsilon\) be a standard convolution kernel, and set \(u_\varepsilon := u * \rho_\varepsilon\). By the Lipschitz regularity assumed of \(u\) on \(U\), for sufficiently small \(\varepsilon > 0\) the matrix \(\smash{(2-p)\, {\partial_{i} u_\varepsilon\partial_{j} u_\varepsilon} /\vert du_\varepsilon\vert^2 - g_{ij}}\) stays uniformly positive definite and \(\vert d u_\varepsilon\vert > 0\) on the support of \(\phi\). Recalling 10 , the \(p\)-d’Alembertian of \(u_\varepsilon\) is \[\begin{align} \label{Eq:dAlem} \square_pu_\varepsilon = \vert du_\varepsilon\vert^{p-2}\left[(2-p)g^{ik}g^{jl}\frac{\partial_{k} u_\varepsilon\partial_{l} u_\varepsilon}{\vert du_\varepsilon\vert^2}-g^{ij}\right]\left[\partial_i\partial_ju_\varepsilon - \Gamma_{ij}^k\,\partial_ku_\varepsilon\right]. \end{align}\tag{16}\] Since \(u_\varepsilon\) is smooth, we can transform the right-hand side into its divergence form and use integration by parts to obtain \[\begin{align} \label{Eq:IBPf} \int_M (g^{ij}\,\partial_i\phi\,\frac{\partial_i u_\varepsilon}{\vert du_\varepsilon\vert^{2-p}})\,\sqrt{\vert g\vert}\,d x = \int_M (\phi\,\square_pu_\varepsilon)\,\sqrt{\vert g\vert}\,dx. \end{align}\tag{17}\] On the other hand, convolution commutes with differentiation; in conjunction with the nonnegativity of \((D^2u)^\perp\) this entails \[\begin{align} \partial_i\partial_ju_\varepsilon = \rho_\varepsilon * (D^2u)_{ij} \geq \rho_\varepsilon * (D^2 u)_{ij}^{\mathrm{a.c.}}. \end{align}\] Here \((D^2 u)^{\mathrm{a.c.}}\) is the \(dx\)-absolutely continuous part of \(D^2u\), and the last inequality is understood in the sense of positive-semidefiniteness of matrix-valued distributions. Since the Hessian \(\partial_i\partial_j u_\varepsilon\) is multiplied by a symmetric and positive-definite matrix in 16 which admits a square-root, the previous observation combines with 17 and the nonnegativity of \(\phi\) to give \[\begin{align} \int_M (g^{ij}\, \partial_i\phi\,\frac{\partial_i u_\varepsilon}{\vert du_\varepsilon\vert^{2-p}})\,\sqrt{\vert g\vert}\,d x \geq \int_V (\phi\,\square_pu_\varepsilon)\,\sqrt{\vert g\vert}\,dx. \end{align}\] Here \(V\) is the open subset of \(U\) where \(u\) is smooth. Its complement relative to \(U\) is negligible since we assumed the past timelike cut locus of \(o\) has measure zero in \(U\).

As \(\varepsilon \to 0\), the left-hand side converges to \[\begin{align} \int_M g^{ij}\,\partial_i\phi\,\frac{\partial_i u}{\vert du\vert^{2-p}}\,\sqrt{\vert g\vert}\,d x \end{align}\] by the dominated convergence theorem, since \(\phi \in C^1_0(U).\) By the same argument, the term containing the Christoffel symbols in 16 in the integral on the right-hand side converges. Lastly, since \(u\) differs from a convex function by the addition of a smooth function \(v\) whose Hessian is bounded on the support of \(\phi\), Fatou’s lemma applies and yields the following limiting estimate for the right-hand side of the previous inequality: \[\begin{align} \liminf_{\varepsilon \to 0} \int_V (\phi\,\square_pu_\varepsilon)\,\sqrt{\vert g\vert}\,dx \geq\int_V(\phi\,\square_pu)\,\sqrt{\vert g\vert}\,dx. \end{align}\]

Finally, by Eschenburg’s smooth d’Alembert comparison [14] (note the different sign conventions on the metric tensor) and the definition of \(u\), on the given set \(V\) its \(p\)-d’Alembertian is bounded from above by \((n-1)/\ell(\cdot,o)\). This implies \[\begin{align} \int_M g^{ij}\,\partial_i\phi\,\frac{\partial_i u_\varepsilon}{\vert du_\varepsilon\vert^{2-p}}\,\sqrt{\vert g\vert}\,d x \geq -\int_M \frac{\phi(n-1)}{l(\cdot,o)}\,\sqrt{\vert g\vert}\,dx, \end{align}\] as desired. ◻

Proof. The reverse triangle inequality 4 shows \(I^-(\gamma(r))\) increases with \(r\); since \(b^+_r(x) = +\infty\) unless \(x \in I^-(\gamma(r))\), we have \(V \subset \cup_{r>0} I^-(\gamma(r))\). Using normal coordinates around \(\gamma(r)\) shows \(b^+_r(\cdot)=const(r)-\ell(\cdot,\gamma(r))\) to be smooth on \(V\), except on the closure of the timelike cutlocus.

In case (a), the timelike cutlocus of \(\gamma(r)\) intersects \(I^-(\gamma(r))\) in a relatively closed set [9] which has of zero volume as a consequence of Theorem 3.5 and the approximate second-differentiability a.e.of the semiconvex function \(\ell(\cdot, \gamma(r)) = const(r)-b_r^+(\cdot)\) described following Definition 3.8 (ibid). Proposition 5 then applies to the approximate Busemann function \(b_r^+\) for fixed \(r>0\) and yields \[\label{weak32p-comparison} \int_{M} \left[\frac{(n-1)\phi}{\ell(\cdot,\gamma(r))} - g\Big(d\phi, \frac{db^+_r}{|db^+_r|^{2-p}}\Big)\right] d{\rm vol}_g\ge 0\tag{18}\] for every nonnegative \(\phi \in C^1_0(I^-(\gamma(r)))\). The convergence provided by Corollary 1 allows us to take \(r \to \infty\) in 18 to get \(p\)-superharmonicity of \(b^+\), semiconcavity and \(|db^+|=1\) on any open subset \(X\) with compact closure in \(V\) by Lebesgue’s dominated convergence theorem, where the monotone limit \(\lim_{r\to \infty}\ell(\cdot,\gamma(r))= \infty\) and the compact support of \(\phi \in C^1_0(X)\) have been used. This concludes case (a).

Case (b) will follow similarly once we have verified that the timelike cutlocus of \(\gamma(r)\) intersects \(I^-(\gamma(r))\) in a relatively closed set of zero volume for each \(r \ge R\). Although Theorem 3.5 (ibid) is stated under hypothesis (a), inspection of its proof reveals it applies equally well to (b) as soon as \(r \ge R\). This is not true of Lemma 2.3 (ibid) however, so we must find another argument to show the intersection in question is relatively closed. Recall the (past) timelike cut locus is contained in the graph of \(G(v) := \exp s(v)v \in M\) over the past unit observer bundle \(T^-_{1}M := \{v \in TM \mid H(-v)=1 \}\), where \(s:T^-_{1}M \longrightarrow [0,\infty]\) is upper semicontinuous according to [2] in case (b). We shall complete the proof by arguing that \(s\) is also lower semicontinuous on the intersection of \(T^-_1\) with \(\exp_{\gamma(r)}^{-1} V \subset T_{\gamma(r)} M\) provided \(r \ge R\), hence \(G\) is continuous there. In case (a) this would follow from [2]. However, the proof of that proposition reveals global hyperbolicity is used only to guarantee the existence of a maximizing geodesic linking \(\gamma(r)\) to each \(x \in V\), which the present context is guaranteed by hypothesis. Case (b) is therefore resolved. ◻

Proof. Proposition 4, Corollary 2 and Theorem 2(ii) combine with Corollary 1 and time-reversal symmetry to show \(-b^-\) and \(b^+\) are \(p\)-superharmonic, semiconcave, and have unit timelike gradients, past-directed in case of \(-b^-\) and future-directed in case of \(b^+\). Thus \(u:=b^+- b^-\) and \(b(t) := b^- + tu\) with \(0 \le \phi \in C^1_0(U)\) yield \[\begin{align} 0 & \le- \int_M d{\rm vol}_g\int_0^1 \frac{d}{dt} g(d\phi, |db|^{p-2} db) dt \\ &= -\int_U d{\rm vol}_g\int_0^1 |db|^{p- 2} g(d\phi, [(p-2)\frac{db}{|db|} \otimes \frac{\nabla b}{|\nabla b|}+I] d u) dt \\ &= +\int_\Omega dx \sqrt{|g|} {\partial}_i\phi {\partial}_j u \int_0^1 |db|^{p- 2}\left[ (2-p) \frac{{\partial}^i b {\partial}^jb}{|d b|^2} - g^{ij} \right] dt \end{align}\] where the last line is expressed in a coordinate chart; we assume \(\mathop{\rm spt}\phi\) is supported in such a chart without loss. Viewing the coefficients as ‘frozen’ shows \(u\) is a weak supersolution \(Lu\ge0\) of the linear operator given in divergence form 12 –13 by \[\begin{align} \label{A32matrix} a^{ij}(x) &= {\sqrt{|g|}}\int_0^1 |db|^{p- 2}\left[ (2-p) \frac{{\partial}^i b {\partial}^jb}{|d b|^2} - g^{ij} \right] dt \end{align}\tag{19}\] Choosing Fermi coordinates near the line \(\gamma\), our signature \((+, -, \dots, -)\) of \(g\) with \(0 \ne p<1\) make it easy to check bounded measurability and uniform ellipticity 14 of these coefficients, taking \(U\) smaller if necessary: along the line \(\gamma(r)\) where \(db^\pm =d\gamma/dr\), the expression in square brackets becomes the diagonal matrix \(diag(1-p,1,\ldots,1)\); near \(\gamma\), semiconcavity ensures that the gradients \(db^\pm\) are not very different from \(d\gamma/dr\) either, Theorems 24.4 and 25.1 of [47]. On the other hand, \(u \ge 0\) throughout \(U\) and vanishes on \(\gamma\), according to 7 . Thus \(u=0\) throughout a neighbourhood \(U\) of \(\gamma\) by the strong maximum principle, Theorem 8.19 of [32]. Now \(b^+=b^-\) is both semiconvex and semiconcave, hence \(b^+ \in C^{1,1}_{loc}(U)\); similarly \(b^+=b^-\) is both \(p\)-super and \(p\)-subharmonic, hence \(p\)-harmonic. ◻

Proof. Evaluating the left hand side of ?? using the chain rule (and the fact that \(v\) derivatives of \(H\) all commute with each other since the cotangent space at each point is flat, while \(x\) derivatives of \(H\) vanish due to the form of our Hamiltonian) yields \[\begin{align} &\nabla_i(H^{ij}H^ku_{jk}) - H^k \nabla_k(H^{ij} u_{ji}) \\=&H^{ijl}H^k(u_{il}u_{jk}-u_{kl}u_{ji}) + H^{ij}u_{jk} H^{kl} u_{il} +H^{ij} H^k (u_{ijk}-u_{kji}) \end{align}\] Here \(u_{ijk} := \nabla_i \nabla_j \nabla_k u\). The terms involving third derivatives of \(H\) cancel each other (since superscripts on \(H\) can be freely permuted and the Levi-Civita connection is torsion free). It remains to see only that the terms involving third derivatives of \(u\) constitute the Ricci curvature term on the right hand side of ?? . But this follows by combining consequences \[\begin{align} \tag{21} H^i &= (\nabla^i u) f'_{|{d}u|^2/2} \qquad {\rm and} \qquad \\H^{ij} &= (\nabla^i u) (\nabla^j u) f''_{|{d}u|^2/2} + g^{ij} f'_{|{d}u|^2/2} \tag{22} \end{align}\] of the structure of our Hamiltonian with the defining property (and antisymmetry) of the Riemann tensor \[\begin{align} u_{ikj}-u_{kij} = {R_{ikj}}^l u_l. \end{align}\] ◻

Proof. Specializing ?? to the Hamiltonian 1 with \(0\ne p<1\), under the strong energy condition ?? , Lemma 2 yields \[\begin{align} \nonumber & g(|du|^{p-2} du , d (\square_p u)) -\nabla \cdot ((D^2 H) d (|du|^p/p)) \\ \label{itbjysum} &=\mathrm{Tr}\left[ \sqrt{D^2H} (\nabla^2 u) D^2 H (\nabla^2 u) \sqrt{D^2H}\right] + |du|^{2p-4} R(du, du), \\& \ge \mathrm{Tr}\left[ \sqrt{D^2H} (\nabla^2 u) (D^2 H) (\nabla^2 u) \sqrt{D^2H}\right] \nonumber \end{align}\tag{23}\] where we have used convexity of the Hamiltonian to take the matrix square-root: strict positive-definiteness of \(D^2 H\) on the timelike future cone was shown in Lemma 3.1 of McCann [9]. Proposition 6 provides a neighbourhood \(U\) of the line \(\gamma\) such that \(b^+ \in C^{1,1}_{loc}(U)\) satisfies \(|db^+|=1\) and \(\square_p b^+=0\) a.e. on \(U\). If \(b^+ \in C^3(U)\), the left-hand side of [Bochner3] vanishes when \(u=b^+\); in this case we conclude \(\sqrt{D^2H} (\nabla^2 b^+) \sqrt{D^2H}=0\) throughout \(U\), and positive-definiteness of \(D^2 H\) yields the desired linearity \(\nabla_i \nabla_j b^+=0\) of \(b^+\).

If instead \(b^+ \in C^{1,1}_{loc}(U)=W^{2,\infty}_{loc}(U)\), there exists a sequence \(u_{\epsilon}\in C^\infty(U)\) with \(|du_{\epsilon}| \ge 1-{\epsilon}\) and \(\|u_{\epsilon}-b^+\|_{W^{2,r}(X)} \to 0\) for each \(r\in[1,\infty)\) and compact \(X \subset U\); moreover \(\nabla^2 u_{\epsilon}\to \nabla^2 b^+\) (hence \(\square_p u_{\epsilon}\to \square_p b^+\)) \({\rm vol}_g\)-a.e.[52]. For each test function \(0 \le \phi \in C^1_0(U)\), evaluating [Bochner3] at \(u_{\epsilon}\) and integrating against \(\phi\) yields \[\begin{align} &\int_U [ (\square_p u_{\epsilon})\nabla \cdot (\phi DH_{\epsilon}) - D^2 H_{\epsilon}(d\phi,d(H_{\epsilon})) ] d{\rm vol}_g \\ &\ge \int_U \phi \mathrm{Tr}\left[ \sqrt{D^2H_{\epsilon}} (\nabla^2 u_{\epsilon}) (D^2 H_{\epsilon}) (\nabla^2 u_{\epsilon}) \sqrt{D^2H_{\epsilon}}\right] d{\rm vol}_g \end{align}\] where \(D^2H_{\epsilon}:=D^2 H|_{du_{\epsilon}}\) and similarly \(DH_{\epsilon}= DH|_{du_{\epsilon}}\) and \(H_{\epsilon}=H|_{du_{\epsilon}}\). Setting \(u_0 := b^+\) and \(X =\mathop{\rm spt}\phi\) yields \(D^2 H_{\epsilon}\to D^2 H_0\) and \(\nabla \cdot (\phi DH_\epsilon) \to \nabla \cdot (\phi DH_0)\) in \(L^{r}(X)\) for all \(r \in [1,\infty)\). Since the sequences \(d H_{\epsilon}\to d H_0=0\) and \(\square_p u_{\epsilon}\to \square_p u_0=0\) and \(\nabla^2 u_{\epsilon}\to \nabla^2 u_0\) converge \({\rm vol}_g\)-a.e. on \(X\) and are bounded, they also converge in \(L^r(X)\) for all \(r \in [1,\infty)\). Choosing \(r=2\), the \({\epsilon}\to 0\) limit yields \[\begin{align} 0 \ge \int_U \phi \mathrm{Tr}\left[ \sqrt{D^2H_0} (\nabla^2 u_0) (D^2 H_0) (\nabla^2 u_0) \sqrt{D^2H_0}\right] d{\rm vol}_g. \end{align}\] Since \(b^+ \in C^{1,1}_{loc}(U)\), the covariant Hessian \(\nabla^2 b^+\) is absolutely continuous with respect to \({\rm vol}_g\); positive-definiteness of \(D^2H_0\) implies \(\nabla^2 b^+\) vanishes \({\rm vol}_g\)-a.e. — hence everywhere — on \(X\). Arbitrariness of \(0 \le \phi \in C^1_0(U)\) concludes the proof that \(b^+\) is smooth and linear throughout \(U\). ◻

Proof in case (b). Taking the neighbourhood \(U\) of \(\gamma\) from Corollary [C:linear] — on which the Hessian of \(b^+ \in C^{\infty}(U)\) vanishes — smaller if necessary, ensures each \(x \in S:=S_0 \cap U\) forms the base \(x=\sigma^\pm(0)\) of forward and backward asymptotes \(\sigma^+:[0,s_+)\longrightarrow M\) and \(\sigma^-:(s_-,0] \longrightarrow M\) to \(\gamma\) (with \(s_+=\infty=-s_-\) in case (b)), both proper-time reparameterized to be future-directed; this follows by using Lemmas 2.1–2.4 of Galloway and Horta [8] to extract timelike subsequential limits \(r\to \infty\) of the maximizing segments from \(x\) to \(\gamma(\pm r)\) provided for \(r\) sufficiently large by our Theorem 2(ii)-(iii). Together they form a future-directed geodesic \(\sigma\), potentially broken at \(\sigma(0)\in S\). Proposition 2.6 of the same reference shows \(b^+(\sigma(r))=r\) for all \(r \in [0,s_+)\); then \[\begin{align} b^-(\sigma(r)) &= b^-(\sigma(r)) - b^-(\sigma(0)) \\& \ge r = b^+(\sigma(r)) \end{align}\] couples with 7 \(b^+ \ge b^-\) to yield \(b^-(\sigma(r))=r\) in the same range of \(r \in [0,s_+)\). Combining the preceding argument with time-reversal symmetry shows both Busemann functions increase along the full asymptote at unit rate: \(b^\pm(\sigma(r))=r\) for all \(r \in (s_-,s_+)\). Because \[b^+(y)-b^+(x) \ge \ell(x,y)\] for all \(x \ll y\), with equality at \((x,y)=(\sigma(r),\sigma(s))\) for all \(r<s\in (s_-,s_+)\), it follows that (i) that \(\sigma\) is a timelike maximizer; (ii) its tangent \(\sigma'(r)=N(\sigma(r))\) agrees with the direction \(N=db^+\) of slowest increase of \(b^+\) on \(U\); and (iii) \(\sigma^\pm\) are \(S_0\)-rays. Thus \(\sigma\) is not broken at \(\sigma(0)\) after all. Moreover, \(\sigma\) is normal to the zero level set \(S \subset S_0\) of \(b^+\), which is spacelike since \(N\) is unit timelike, and smooth by the implicit function theorem.

The arguments above show that \(E(r,x):=\exp_x rN\), with maximal domain of definition \(\mathcal{D} \subset {\mathbf{R}}\times S\), has the following properties: The trajectories \(E(\cdot,x)\) and \(E(\cdot,y)\), for \(x,y \in S\), are timelike lines which do not cross (unless \(x=y\)), contain no conjugate points nor focal points to \(S\), and maximize the time separation to \(S\). In case (a) \(\mathcal{D}\) is open by [2], while in case (b), \(\mathcal{D} = {\mathbf{R}}\times S\) holds automatically. In both cases the implicit function theorem shows \(E\) is a diffeomorphism from \(\mathcal{D} \subset {\mathbf{R}}\times S\) onto a set \(W \subset M\) which is open, and \(E(r,\gamma(0))=\gamma(r)\) for all \(r \in {\mathbf{R}}\).

We have now shown \(b^+=b^-\) throughout \(W\) (in addition to \(U\)). Since \(E\) is a smooth diffeomorphism and \(b^+(E(r,x))=r\) for each \((r,x)\in \mathcal{D}\), the continuous differentiability of \(b^\gamma:=b^\pm\) propagates throughout \(W\). On \(U\), recall \(b^\gamma\) is smooth and its Hessian vanishes, from Corollary [C:linear]. Thus \(N=db^\gamma\) satisfies Killing’s equation in \(U\), hence is parallel throughout \(U\), so that any geodesic in \(U \subset M\) has constant inner product with \(N\). In particular, since \(N=db^\gamma\) is normal to the zero level \(S^\gamma := S_0\cap U =S_0\cap W\) of \(b^\gamma\), we see \(S\) is totally geodesic: its second fundamental form (or shape operator) vanishes.

At this point we specialize to case (b), defering the completion of case (a) to the following section. In case (b) the line \(\sigma\) is complete, so applying the same logic to it as to \(\gamma\) yields a neighbourhood \(V\) of \(\sigma\) on which the forward Busemann function \(b^\sigma\) is smooth and has vanishing Hessian. Thus its zero set \(S^\sigma := \{x\in V \mid b^\sigma(x)=0 \}\) is also totally geodesic in \(V\). Taking \(V\) smaller if necessary again ensures each \(x \in S^\sigma\) lies on a complete line \(\beta^x:(-\infty,\infty) \longrightarrow M\) with \(\beta^x(0)=x\) which is (future and past) asymptotic to \(\sigma\). Since both \(S^\gamma\) and \(S^\sigma\) are orthogonal to \(\sigma\) and are totally geodesic, we conclude they coincide in the neighbourhood \(V \cap W\) of \(\sigma(0)\). Now the asymptotes to \(\sigma\) and to \(\gamma\) both intersect \(S\) orthogonally in \(W \cap V\), so for each \(x \in S\cap V\) these asymptotes (\(E(\cdot,x)\) to \(\gamma\) and \(\beta^x(\cdot)\) to \(\sigma\)) also coincide. Moreover both \(b^\gamma\) and \(b^\sigma\) increase at rate one along these asymptotes, hence \(b^\gamma=b^\sigma\) throughout \(W\cap V\). Thus \(b^\gamma\) inherits the linearity of \(b^\sigma\): its Hessian vanishes throughout \(W \cap V\). Since the asymptote \(\sigma\) of \(\gamma\) had arbitrary base \(\sigma(0) \in S\), we conclude the Hessian of \(b^\gamma\) vanishes globally on \(W\).

We have now shown the flow map \(F(r,x)\) of \(N=db^\gamma\) satisfying \(\frac{dF}{dr} = N(F(r,x))\) and \(F(0,x)=x\) coincides with \(E\) on \({\mathbf{R}}\times S\). The fact that the Hessian of \(b^\gamma\) vanishes means \(N=db^\gamma\) is parallel throughout \(W\) and satisfies Killing’s equation. Thus \(F(r,\cdot):W \longrightarrow W\) pulls-back the metric \(g\) to itself for each \(r \in {\mathbf{R}}\), which shows \(E(r,S)\) to be isometric to \(S=E(0,S)\). Along the totally geodesic surface \(S\) normal to \(N\), the metric \(g\) therefore splits into the direct sum of its restriction \(-h\) to \(S\) plus \(dr^2\) in the orthogonal direction \(N\). Thus \(E\) gives the desired local isometry between \((\mathcal{D}={\mathbf{R}}\times S, dr^2-h)\) and \((W,g)\). ◻

Proof. This follows from Theorem 10, the definition of the product metric \(dr^2-h\), and of \(\ell\) from 2 . ◻

Proof. The definition of asymptote asserts \(\alpha\) is a (subsequential) limit curve of a sequence of maximizing geodesics \(\sigma_r:[0,s_r] \longrightarrow M\) with \(\sigma_r(0)=\alpha(0) \in I^+(S) \cap I^-(\gamma)\) and \(\sigma_r(s_r) = \gamma(r)\) as \(r \to \infty\). In other words, as \(r\to\infty\) along a subsequence, \(\tilde{\sigma}_r\) converges uniformly to \(\tilde{\alpha}\) on compact subsets of \([0,\infty)\), where \(\tilde{\sigma}_r\) and \(\tilde{\alpha}\) denote the \(\tilde{g}\)-arclength reparameterizations of \(\sigma\) and \(\alpha\) respectively. Such an asymptote is a ray by [8], hence \(\tilde{\alpha}\) can be extended to \((-\infty,\infty)\) and affinely reparameterized as a future- and past-inextendible geodesic \(\alpha:(a_-,a_+)\longrightarrow M\) for some \(a_- \in [-\infty,0)\), and \(a_+\in (0,\infty]\) with \(\alpha(0)=\tilde{\alpha}(0)\).

Strong causality of \((M,g)\) and [53] imply for \(\epsilon>0\) sufficiently small, that \(\tilde{\alpha}(-\epsilon) \in I^-(S)\) and \(\tilde{\alpha}|_{[-\epsilon,\epsilon]}\) is maximizing. As in the last paragraph of the proof of Proposition 4 (and of Proposition 11), setting \(x =\alpha(0)\) and \(y_r=\tilde{\sigma}_r(\epsilon)\) yields \[b^+_r(\tilde{\alpha}(s)) \le b^+_r(x) + \ell(x,y_r) - \ell(\tilde{\alpha}(s),y_r)\] for all \(|s|<\epsilon\). To derive a contradiction, assume \(\alpha\) is null. Letting \(r \to \infty\) along the relevant subsequence yields \[\begin{align} b^+(\tilde{\alpha}(s)) &\le b^+(x) + \ell(x,\tilde{\alpha}(\epsilon)) - \ell(\tilde{\alpha}(s),\tilde{\alpha}(\epsilon)) \\ &= b^+(\tilde{\alpha}(0)) \end{align}\] by the nullity of \(\tilde{\alpha}\) and our choices of \(\epsilon\) and \(|s|<\epsilon\). Differentiation at \(s=0\) then shows \(db^+|_x(\tilde{\alpha}'(0))=0\), so \(db^+|_x\) is vanishing, spacelike or null. On the other hand, the Lipschitz continuity hypothesized of \(b^+\) around \(x=\alpha(0)\) yields \(db^+|_x\) timelike (and future-directed, with Lorentzian magnitude at least \(1\)) as in Remark 3. This contradiction forces \(\alpha\) to be timelike as claimed. ◻

Proof. Let \(\alpha:[0,a_+)\longrightarrow M\) be a (future inextendible, proper-time parameterized) asymptote to \(\gamma\) with \(b^+\) continuously differentiable in a neighbourhood of \(x_0=\alpha(a)\) for some \(a\in [0,a_+)\). Since [8] asserts \(b^+\) increases at its minimal rate along \(\alpha\), it follows from 11 that \(\alpha'(a)=db^+(x_0)\). Fixing \(0<\epsilon <a_+-a\) yields \(y_0:=\alpha(a+\epsilon) = \exp_{x_0} \epsilon db^+(x_0)\). Since the asymptote is timelike and maximizing, [2] combines with [9] to yield a compact neighbourhood \(V \subset TM\) of \((x_0,\epsilon db^+(x_0))\) in the timelike future bundle on which the exponential map is defined and \(\ell\) is smooth on \(\exp V\). Take \(C\) large enough that \(C \tilde{g}\) dominates the \(\tilde{g}\)-covariant Hessian (with respect to \(x\)) of \(v(x):=-\ell(x,y)\) for all \((x,y) \in \exp V\). We claim \(C\) is the desired semiconcavity constant.

Let \(X \times Y \subset \exp V\) be a neighbourhood of \((x_0,y_0)\), where \(\exp\) here denotes the map taking a vector \(v_p\) to \((p,\exp_p(v)) \in M^2\). Since \(b^+(x_0)<\infty\), taking \(R\) sufficiently large ensures \(x_0 \in I^-(\gamma(R))\) (by the push-up property). Taking \(X\) smaller if necessary guarantees \(X \subset W \cap I^-(\gamma(R))\) as well. For each \(x \in X\) and \(r \ge R+1\), global hyperbolicity provides a proper-time parameterized maximizing geodesic segment \(\sigma_r:[0,s_r]\longrightarrow M\) from \(\sigma_r(0)=x\) to \(\sigma_r(s_r)=\gamma(r)\). The reverse triangle inequality shows \(s_r \to \infty\), since \(\ell(\gamma(R),\gamma(r)) \to \infty\) as \(r \to \infty\). Set \(y_r=\sigma_r(\epsilon)\). By Lemmas 2.1 (the limit curve theorem) and 2.4 of [8], one can extract an asymptote \(\sigma\) to \(\gamma\) as a subsequential limit curve of \(\sigma_r\) as \(r \to \infty\). Since \(x \in W\) and \(b^+ \in C^1(W)\), Lemma 3 asserts this asymptote is timelike. Now [8] again asserts \(b^+\) increases at its minimal rate along \(\sigma\), thus \(\sigma(s)=\exp_x sdb^+(x)\) by the same logic as above, so the full sequence \(\sigma_r \to \sigma\) on \([0,a+\epsilon]\) and \(y_r \to y\) as \(r \to \infty\). Our hypothesis \(b^+ \in C^1\) yields \(y \to y_0\) as \(x \to x_0\), so taking \(X\) smaller and compact if necessary ensures \(y \in Y\) for all \(x \in X\) and all \(r \ge R+1\). Since \(y_r \to y\) as \(r \to \infty\), taking \(R\) larger (independently of \(x\) within the compact set \(X\)) then ensures \(y_r \in Y\).

For each \(x \in X\) and \(r \ge R\), Lemma 1 asserts \(u(x') := b^+_r(x) + \ell(x,y_r) - \ell(x',y_r) \ge b^+_r(x')\) for all \(x'\) near \(x\). Since equality holds at \(x'=x\), this means \(u(x')\) supports \(b^+_r\) from above at \(x\). Thus \(b^+_r\) inherits the asserted semiconcavity constant \(C\) chosen above from \(u\) at \(x\). ◻

Proof. Proposition 11 provides the equi-semiconcavity of \(\{b^+_r\}_{r \ge R}\) necessary for Corollary 2 to imply \(|db^+|=1\) and semiconcavity and \(p\)-superharmonicity of \(b^+\) on the interior of \(X\); global hyperbolicity ensures timelike maximizing geodesics connect each \(x \in X \subset I^-(\gamma(R))\) to \(\gamma(r)\) for \(r \ge R\). ◻

Proof of Theorem 10 in case (a). Recall from the proof of case (b) that in both cases we had identified a neighbourhood \(U\) of \(\gamma\) on which the Busemann functions \(b=b^\pm\) coincide and have vanishing Hessian (hence \(db\) is a parallel Killing vector field on \(U\)), and a neighbourhood \(W\) of \(\gamma\) foliated by timelike lines \(r\in (r^+_x,r^-_x) \mapsto E(r,x)\) which are future- and past-asymptotic to \(\gamma\) for each \(x \in S\), where \(S=U\cap S_0=W \cap S_0\) is spacelike and totally geodesic with timelike unit normal \(N=db\) and \(S_0 = \{ b=0\}\). In fact \(W\) was the image of an open set \(\mathcal{D} \subset {\mathbf{R}}\times S\) under the smooth diffeomorphism \(E(r,s)=\exp_x r db(x)\); moreover, \(b^\pm(E(r,x))=r\) on \(\mathcal{D}\), so \(b=b^\pm\) coincide and are smooth on \(W\). We now argue their Hessians vanish there.

Corollary 5 and its time-reversal show \(\pm b^\pm\) to be \(p\)-superharmonic on \(W\) and \(|db^\pm|=1\), thus \(b=b^\pm\) is \(p\)-harmonic and satisfies the conclusions of Proposition 6 on \(W\). Corollary [C:linear] then asserts that the Hessian of \(b\) vanishes on \(W\). Thus \(db\) is a parallel Killing vector field throughout \(W\) (as was already known on \(U\)). This shows \(E\) gives a local isometry between the metrics \(dr^2 -h\) on \(\mathcal{D}\) and \(g\) on \(W\), where \(-h\) is the restriction of \(g\) to \(S\).

Taking \(U\) (and thus \(W\)) smaller if necessary ensures \(S\) is a geodesic ball centered at \(\gamma(0)\). It remains to deduce that \(E(\cdot,x)\) is complete for each \(x \in S\), so that \(\mathcal{D} = {\mathbf{R}}\times S\). This is shown as in Galloway [15], but we recall the argument for the convenience of the reader; we only argue future-completeness. Let \(\hat{\sigma}:[0,R) \to S\) be any radial geodesic starting from \(\hat{\sigma}(0)=\gamma(0)\), where \(R\) is the radius of the ball \(S\). Let \(\alpha_s(\cdot):=E(\cdot,\hat{\sigma}(s))|_{[0,l_s)}\) be future-inextendible, and \(l_s =r^+_{\hat{\sigma}(s)}\in (0,\infty]\) its Lorentzian arclength. Fix any \(r > R\). We will now show that \[\label{Galloway39s3242} l_s > r - s\tag{24}\] for all \(s \in [0,R)\), thus (by arbitrariness of \(r>R\) and \(\hat{\sigma}\)) establishing the claimed future completeness of the asymptotic lines to \(\gamma\) based in \(S\). Denote by \(A\) the set of \(t \in [0,R)\) such that 24 holds for all \(s \in [0,t]\). Clearly, \(A\) is an interval containing \(0\), since \(\alpha_0 = \gamma|_{[0,\infty)}\). Let \(a:=\sup A\). We deduce \(a=R\) by showing the non-empty interval \(A\) is both relatively open and closed in \([0,R)\). First observe that since \(\mathcal{D}\) is open (or equivalently, recalling lower semicontinuity of \(l_s = r^+_{\hat{\sigma}(s)}\) from [2]), it follows that \(A\) is a relatively open subset of \([0,R)\). On the other hand, we will derive a contradiction by showing \(a< R\) implies \(a \in A\). We may also assume \(a > 0\) since \(0 \in A\) has already been checked. By definition, \(l_s > r - s\) for any \(s \in [0,a)\). The geometry we have established on \(W\) shows \(\eta(u):=E(r-u,\hat{\sigma}(u))\) to be a past-directed null geodesic \(\eta:[0,a) \to M\) from \(\eta(0) = \gamma(r)\). We claim that \(l_a > r - a\). If not, then \(l_s > r - s > r - a \geq l_a\) for any \(s \in [0,a)\). Thus, \(\alpha_s(t)\) is well-defined for \(s \in [0,a)\) and \(t \in [0,l_a)\), and \(\alpha_a(t) = \lim_{s \to a} \alpha_s(t)\) by continuity of \(E\). It follows that for \(s \in [0,a)\), \[\alpha_s(t) \ll \alpha_s(r - s) = E(r-s,\hat{\sigma}(s)) = \eta(s) \leq \gamma(r).\] Taking \(s \to a\) above and using the closedness of the causal relation guaranteed by global hyperbolicity (a), we conclude that \(\alpha_a \subset J^+(\hat{\sigma}(a)) \cap J^-(\gamma(r))\), a contradiction to non-total imprisonment. Thus \(l_a > r -a\) and hence \(a \in A\). To avoid this contradiction, \(a = R\) and 24 holds for all \(s \in [0,R)\), to establish the claim. ◻

Proof of Theorem 1. Let \(W \subset M\) be the largest connected open subset (ordered by inclusion) on which the conclusion of Theorem 10 holds, meaning \(E(r,x) := \exp_x r db^+\) gives a local isometry from \(({\mathbf{R}}\times S, dr^2 -h)\) onto its image \(W\) in \((M,g)\), and \(r \in {\mathbf{R}}\mapsto E(r,x)\) is an \(S_0\)-line for each \(x \in S=W \cap S_0\) (and \(\gamma(\cdot)=E(\cdot,\gamma(0))\)), and \(h\) is the restriction of \(-g\) to \(S\). Such a subset exists by Zorn’s lemma, and is non-empty by Theorem 10. We claim \(W\) is a connected component of \(M\).

Whenever \(\hat{\sigma}:[0,s_0] \longrightarrow S\) is an \(h\)-geodesic in \(S\), then \(F(r,s) = E(r,\hat{\sigma}(s))\) is a flat strip, so its boundaries are strongly parallel. Lemma 4 shows the Busemann functions associated with the lines \(E(\cdot,x)\) and \(E(\cdot,y)\) through \(x=\hat{\sigma}(0)\) and \(y =\hat{\sigma}(s_0)\) coincide. Since \(W\) is an open and connected Lorentzian product of \(({\mathbf{R}},dr^2)\) with \((S,h)\), it follows that the hypersurface \(S\) is totally geodesic and path-connected. Any path in \(S\) from \(\gamma(0)\) to \(x \in S\) can therefore be replaced by a broken geodesic consisting of arbitrarily short geodesic segments, so iterating the argument above yields a finite sequence of lines starting with \(\gamma\) and ending with \(\beta(\cdot) = E(\cdot, x)\) such that each adjacent pair of lines in the sequence is strongly parallel. Thus \(I(\gamma) = I(\beta)\) and the Busemann functions associated with \(\gamma\) and \(\beta\) coincide.

To derive a contradiction, suppose \(W\) has a boundary point \(y \in {\partial}W\). Fix a coordinate chart around \(y\). Every Euclidean ball of sufficiently small radius in these coordinates centered near \(y\) will be Lorentzian-geodesically convex. Choose such a ball \(B_{2\delta}(y)\) centered at \(y\) and then \(x \in W\cap B_\delta(y)\) sufficiently close to \(y\) that the largest Euclidean ball \(B_\epsilon(x)\) in \(W\) is also Lorentzian-geodesically convex. This maximality of \(\epsilon < 2\delta\) implies there exists \(z \in {\partial}B_\epsilon(x) \cap {\partial}W\). Moreover, there is a Lorentzian geodesic \(\sigma:[0,s_0] \longrightarrow M\) in \(B_\epsilon(x)\) from \(x=\sigma(0)\) to \(z=\sigma(s_0)\) of the form \(\sigma(s) = ((1-s/s_0)r_0 + sr_1/s_0,\hat{\sigma}(s))\) where \(\hat{\sigma}(s)\) is a (nonconstant) \(h\)-geodesic in \(S\). For each \(s<s_0\), the preceding paragraph shows the line \(E(\cdot,\hat{\sigma}(s))\) through \(\sigma(s)\) shares the same Busemann function as \(\gamma\). The product geometry guarantees that the tangent \(X(s)\) to this line at \(\sigma(s)\) is the parallel translate along \(\sigma\) of the tangent \(X(0)\) to the analogous line through \(x=\sigma(0)\). Letting \(s\to s_0\), [8] provides a subsequence of these lines which converge in the limit curve sense to a line through \(z=\sigma(s_0)\). Since the parallel transport \(X(s_0)\) of \(X(0)\) along \(\sigma\) is tangent to this line, and timelike, we can proper-time reparameterize the limiting line as \(\tilde{\gamma}:(a_-,a_+)\longrightarrow M\). By the time-translation symmetry of \(W\) we can assume \(r_1=0\), and choose \(z=\tilde{\gamma}(0)\) to lie in the closure of \(S\).

In case (b) the line \(\tilde{\gamma}\) is complete: \(a_+=\infty=-a_-\). Denote its forward Busemann function by \(\tilde{b}^+\). Theorem 10 provides a neighbourhood \(\tilde{W}\) of \(\tilde{\gamma}\) on which \(\tilde{E}(r,x) := \exp_x r d\tilde{b}\) gives a local isometry \(\tilde{E}:({\mathbf{R}}\times \tilde{S}, dr^2 - \tilde{h}) \longrightarrow (M,g)\) such that \(r \in {\mathbf{R}}\mapsto \tilde{E}(r,x)\) is an \(\tilde{S}_0\)-line for each \(x \in \tilde{S}=\tilde{W} \cap \tilde{S}_0\) (and \(\tilde{\gamma}(\cdot)=\tilde{E}(\cdot,\tilde{\gamma}(0))\)), and \(\tilde{h}\) is the restriction of \(-g\) to \(\tilde{S}\). Since the product geometry shows both \(S_0\) and \(\tilde{S}_0\) are totally geodesic and orthogonal to \(X(s_0)\), they must coincide on the nonempty set \(W \cap \tilde{W}\) — as must \(E\) and \(\tilde{E}\). Since \(\tilde{F}(r,s) = \exp_{\sigma(s)} r X(s)\) is a flat strip bounded by \(\beta\) and \(\tilde{\gamma}\), Lemma 4 shows \(b^+=\tilde{b}^+\) hence \(S_0=\tilde{S}_0\). Note for \(w \in S\) and \(\tilde{w} \in \tilde{S}\), the lines \(E(\cdot,w)\) and \(\tilde{E}(\cdot,\tilde{w})\) cannot cross (unless they coincide), since both are assumed to maximize time to \(S_0\). Thus \(W \cup \tilde{W}\) provides an enlargement of \(W\) on which the conclusion of Theorem 10 holds, contradicting the assumed maximality of \(W\). This contradiction forces \({\partial}W\) to be empty; connectedness of \(M\) yields \(M=W\), and the splitting becomes global by Corollary 4. Ricci nonnegativity of \((S,h)\) follows from the strong energy condition ?? by the tensorization of the Ricci tensor in product geometries, as in [54] with trivial warping factor \(f=1\).

The logic and conclusion of the preceding paragraph will apply to case (a) also as soon as completeness of the timelike line \(\tilde{\gamma}\) is established. We’ll show future-completeness \(a_+=\infty\) as in [15]; past-completeness \(a_-=-\infty\) can be shown similarly (or by time-reversal symmetry). Defining \(\tilde{F}(r,s) = \exp_{\sigma(s)} r X(s)\) as above, it remains true that the restriction of \(\tilde{F}\) to \({\mathbf{R}}\times [0,s]\) is a flat strip for each \(s<s_0\). Moreover, for each \(t \in (a_-,a_+)\) there is a sequence \((r_i,s_i) \in {\mathbf{R}}\times [0,s_0)\) such that \(\tilde{F}(r_i,s_i) \to \tilde{\gamma}(t)\). Since \(\tilde{\gamma}(t)\) is separated from \(\tilde{\gamma}(0) \in \bar S\) by time \(t<a_+\) and \(\tilde{F}(r_i,s_i)\) is separated from \(\tilde{\gamma}(0)\) by time not much less than \(r_i + (1-s_i/s_0)r_0\), continuity of the time-separation function \(\ell\) implies \(r_i < a_+ + 1\) for \(i\) sufficiently large. Also \(R>|r_0| + a_++1 + |\hat{\sigma}'(0)|_h\) ensures \(\tilde{F}(R,0)\) lies in the future of \(\tilde{F}(r_i,s_i)\) for all \(i\) sufficiently large, by ?? . Since \(t<a_+<\infty\) was arbitrary, the restriction \(\tilde{\gamma}|_{[0,a_+)}\) — being future-inextendible — is a timelike curve of unbounded \(\tilde{g}\)-length in the compact diamond \(J(\tilde{\gamma}(0),\tilde{F}(R,0))\), contradicting the nontotal imprisonment which global hyperbolicity (a) implies. We therefore conclude future-completeness of \(\tilde{\gamma}\): \(a_+=+\infty\) (and past-completeness \(a_-=-\infty\) similarly).

Apart from (metric) completeness of \((S,h)\), Theorem 1 has now been established. To see completeness of \((S,h)\), let \(x_k \in S\) denote a Cauchy sequence. Then the entire sequence \(\{x_k\}_{k \in {\mathbf{N}}}\) lies in an open \(h\)-ball \(B_r(x_1)\) of radius \(r\) sufficiently large. From 2 it follows that \(x_k\) also lies in the diamond \(J(E(-r,x_1),E(r,x_1))\), which is compact assuming (a) global hyperbolicity. In this case \(x_k\) admits a subsequential limit \(x_\infty\) in \(M\). We claim \(d_h(x_k,x_\infty) \to 0\). This follows from the facts (i) that \(d_h\) metrizes the topology \(S\) inherits from \(M\), and (ii) that whenever a Cauchy sequence has a convergent subsequence then the full sequence also converges (to the same limit).

If instead \((M,g)\) is (b) timelike geodesically complete, we will assume incompleteness of \((S,h)\) to derive a contradiction. In this case the Hopf-Rinow theorem provides an \(h\)-geodesic \(\tau:(s,t) \longrightarrow S\) which is inextendible (say at \(t\)). Lifting \(\tau\) produces a timelike geodesic \(\beta(r):=(r,\tau(r/2))\) in the product metric \(dr^2 -h\), which is future-inextendible at \(r=2t\): the desired contradiction to (b). So \((S,h)\) is complete and Theorem 1 is established. ◻