Proof. Consider the following subset of \(T_{x}M\): \[\begin{align} V:=\{v\in C^{-}_{x}:|v|_{g}=|u_{x}|_{g}\}\;. \end{align}\] We now show that, if conditions \((i)\) and \((ii)\) are satisfied, then \(V\) is a proper subset of the zero set of \(F\) on \(C^{-}_{x}\), contradicting condition \((iii)\).

The set \(V\) contains at least two non-collinear vectors: given an generalized orthonormal basis \(\{e_{i}\}_{i=0}^{n}\) of \(T_{x}M\), the vectors \(v_{1}:=|u_{x}|_{g}e_{0}\) and \(v_{2}:=|u_{x}|_{g}(\cosh{(\varphi)}e_{0}+\sinh{(\varphi)}e_{1})\) are non collinear for \(\varphi\neq0\) and belong to \(V\). The (strict) reversed triangle inequality applied to two non-collinear \(v_{1},v_{2}\in V\) implies that \(V\) is not a convex set: \[\begin{align} |\frac{1}{2}v_{1}+\frac{1}{2}v_{2}|_{g}>\frac{1}{2}|v_{1}|_{g}+\frac{1}{2}|v_{2}|_{g}=|u_{x}|_{g} \implies \frac{v_{1}+v_{2}}{2}\not\in V\;. \end{align}\] However, the zero set of \(F\) is convex, as follows from conditions \((i)\) and \((ii)\). Let \(v_{1}\) and \(v_{2}\) belong to the zero set of \(F\) and \(\lambda\in(0,1)\): \[\begin{align} F(\lambda v_{1}+(1-\lambda)v_{2})\leq \lambda F(v_{1})+(1-\lambda)F({v_{2}})=0\implies F(\lambda v_{1}+(1-\lambda)v_{2})=0\;. \end{align}\] Since \(V\) is a non-convex subset of the zero set of \(F\), which is convex, it is a proper subset. ◻

Proof. In the first place, coercivity of the functional \(F^{\alpha}_{p,q}\) follows from coercivity of \(F_{h,p}\). Applying the Poincaré-Wirtinger inequality (cf. [17]) to the functions in \(\mathcal{T}^{p}\) (so they satisfy the zero-mean condition) gives \[\begin{align} \|t\|_{H^{p}(\Omega)}^{2}&=\|t\|_{L^{2}(\Omega)}^{2}+\|\nabla^{h} t\|_{H^{p-1}(\Omega)}^{2}\leq C\|\nabla^{h} t\|_{L^{2}(\Omega)}^{2}+\|\nabla^{h}t\|_{H^{p-1}(\Omega)}^{2}\leq C\|\nabla^{h}t\|_{H^{p-1}(\Omega)}^{2} \end{align}\] with \(C>0\) a constant depending on \(\Omega\), \(p\) and \(h\). Note that, in this proof, constants might change from one inequality to another but, as is customary, will still be denoted with the same letter. Since \(\Omega\) is compact and \(u\) and \(g\) are smooth \[\begin{align} F^{\alpha}_{p,q}(t)\geq F_{h,p}(t)\geq C\|\nabla^{g} t\|_{H^{p-1}(\Omega)}^{2}-\|u\|_{H^{p-1}(\Omega)}^{2}\geq C\|\nabla^{h} t\|_{H^{p-1}(\Omega)}^{2}-C'\geq C\|t\|_{H^{p}(\Omega)}^{2}-C' \end{align}\] where we used that by compactness of \(\Omega\) and smoothness of \(g\) and \(h\) there exists a constant \(c\) such that \(\|\nabla^{g} t\|_{H^{p-1}(\Omega)}^{2}\geq c\|\nabla^{h} t\|_{H^{p-1}(\Omega)}^{2}\). Finally, \(C,C'>0\) depend also on \(u\) and \(g\). Hence, \(F^{\alpha}_{p,q}\) and \(F_{h,p}\) are coercive.

Secondly, let us address convexity of \(F_{h,p}\) and \(F_{g,q}\). Note that for any \(t_{1},t_{2}\in \mathcal{T}^{p}\), \(t_{1}\neq t_{2}\) implies that \(\nabla^{g}t_{1}\neq\nabla^{g}t_{2}\) (if \(\nabla^{g}t_{1}=\nabla^{g}t_{2}\) a.e., then \(t_{1}=t_{2}+C\) a.e. but \(C=0\) by the zero mean condition). Strict convexity of the \(H^{p-1}(\Omega)\) norm squared entails strict convexity of \(F_{h,p}\).

For the proof of convexity of \(F_{g,q}\), consider arbitrary \(t_{1},t_{2}\in\mathcal{T}^{p}\) with \(t_{1}\neq t_{2}\). Convexity of \(F_{g,q}\) is immediate if the gradient of one of them is null or vanishes on a positive measure subset of \(\Omega\). Assume now that both \(\nabla^{g}t_{1}\) and \(\nabla^{g}t_{2}\) are past-directed timelike a.e.. The reversed triangle inequality gives for any \(\lambda\in(0,1)\) \[\begin{align} \label{eq:reversed95triangle} |\lambda\nabla^{g}t_{1}+(1-\lambda)\nabla^{g}t_{2}|_{g}\geq \lambda|\nabla^{g}t_{1}|_{g}+(1-\lambda)|\nabla^{g}t_{2}|_{g}\;, \end{align}\tag{4}\] which yields strict convexity on \(\mathcal{T}^{p,t}\). First consider the case with \(|\nabla^{g}t_{1}|_{g}\neq|\nabla^{g}t_{2}|_{g}\), \[\begin{align} F_{g,q}(\lambda t_{1}+(1-\lambda)t_{2}) &\leq \int_{\Omega}\frac{1}{(\lambda|\nabla^{g}t_{1}|_{g}+(1-\lambda)|\nabla^{g}t_{2}|_{g})^{2q}}d\mu_{h}\\ &< \lambda\int_{\Omega}\frac{1}{|\nabla^{g}t_{1}|_{g}^{2q}}d\mu_{h} +(1-\lambda)\int\frac{1}{|\nabla^{g}t_{2}|_{g}^{2q}}d\mu_{h}\\ &=\lambda F_{g,q}(t_{1})+(1-\lambda)F_{g,q}(t_{2})\;, \end{align}\] where for the second inequality we used that the function \(f: (0,\infty)\to(0,\infty), x\mapsto \frac{1}{x^{2q}}\) is strictly convex for any \(q\geq1\), so the same holds for \(F_{g,q}\) on \(\mathcal{T}^{p,t}\). On the other hand, if \(|\nabla^{g}t_{1}|_{g}=|\nabla^{g}t_{2}|_{g}\) and \(\nabla^{g}t_{1}\neq\nabla^{g}t_{2}\) on a positive measure subset, then the two gradients are not collinear on this set and one has a strict inequality in 4 which gives again strict convexity of \(F_{g,q}\).

It remains to prove weak lower semicontinuity of the functionals. Let \(i\in\{0,1,\ldots,p-1\}\). The linear map \(t\in H^{p}(\Omega)\mapsto (\nabla^{h})^{i}\nabla^{g}t\in L^{2}(\Omega)\) is bounded and thus continuous, \[\begin{align} &\|(\nabla^{h})^{i}\nabla^{g}t\|^{2}_{L^{2}(\Omega)}\leq C(\|t\|_{L^{2}(\Omega)}^{2}+\sum_{m=0}^{p-1}\|(\nabla^{h})^{m}\nabla^{h}t\|_{L^{2}(\Omega)}^{2})=C\|t\|_{H^{p}(\Omega)}^{2}\;, \end{align}\] so also the affine map \(\psi_{i} : H^{p}(\Omega)\rightarrow L^{2}(\Omega), t\mapsto (\nabla^{h})^{i}(u-\nabla^{g}t)\) is continuous. Since \(F_{h,p}(t)=\sum_{i=0}^{p-1}\|\psi_{i}(t)\|_{L^{2}(\Omega)}^{2}\), also \(F_{h,p}\) is continuous and, since it is convex, sequential weak lower semicontinuity follows (see [28]).

Note that, even though the real valued function \(t\mapsto|\nabla^{g}t|_{g}^{-2q}\) is continuous on \(\mathcal{T}^{p,t}\), this does not imply continuity of \(F_{g,q}\) on \(\mathcal{T}^{p,t}\) and hence the previous argument cannot be used here in order to conclude that the functional is weakly lower semicontinuous. For this purpose, the additional assumption that \(p\geq 2\) is exploited.

Consider a sequence \((t_{n})_{n\in\mathbb{N}}\) in \(\mathcal{T}^{p,t}\) converging weakly in \(H^{p}(\Omega)\) to \(t_{\ast}\in \mathcal{T}^{p,t}\) and a subsequence \((t_{n_{k}})_{k\in\mathbb{N}}\) for which \((F_{g,q}(t_{n_{k}}))_{k\in\mathbb{N}}\) converges to \(\liminf_{n\to \infty}F_{g,q}(t_{n})<\infty\) (if it diverges weak lower semicontinuity of \(F_{g,q}\) is immediate).

By the Rellich-Kondrachov theorem \((p\geq2)\) the Sobolev space \(H^{p}(\Omega)\) is compactly embedded in \(H^{1}(\Omega)\), so (after passing to a further subsequence) \((t_{n_{k}})_{k\in\mathbb{N}}\) converges strongly in \(H^{1}(\Omega)\) to \(t'_{\ast}\in H^{1}(\Omega)\). From uniqueness of the weak limit (strong \(H^{1}(\Omega)\)-convergence implies weak \(H^{1}(\Omega)\)-convergence) it follows that \(t_{\ast}'=t_{\ast}\). In particular, \((\nabla^{g}t_{n_{k}})_{k\in\mathbb{N}}\) converges in the \(L^{2}(\Omega)\)-norm to \(\nabla^{g}t_{\ast}\), which (passing again to a subsequence) implies a.e. convergence and so also \((|\nabla^{g}t_{n_{k}}|^{-2q}_{g})_{k\in\mathbb{N}}\) converges to \(|\nabla^{g}t_{\ast}|^{-2q}_{g}\) a.e. in \(\Omega\) (composition with continuous maps preserves a.e. convergence). Since these are non-negative functions, Fatou’s lemma implies weak lower semicontinuity of \(F_{g,q}\): \[\begin{align} \liminf_{n\to\infty} F_{g,q}(t_{n})=\liminf_{k\to\infty} F_{g,q}(t_{n_{k}})&\geq\int_{\Omega}\liminf_{k\to\infty}\frac{1}{|\nabla^{g}t_{n_{k}}|_{g}^{2q}}d\mu_{h}=F_{g,q}(t_{\ast})\;. \end{align}\] This concludes the proof. ◻

Proof. In the first place, consider the case with \(\alpha>0\). Fix an arbitrary \(q\in \mathbb{N}\). Existence of a minimizer follows from the direct method of the calculus of variations and the assumption that \(p\geq 2\).

Consider a sequence \((t_{n})_{n\in\mathbb{N}}\) in \(\mathcal{T}^{p}\) such that \((F^{\alpha}_{p,q}(t_{n}))_{n\in\mathbb{N}}\) converges to \[\begin{align} F_{\inf}:=\inf_{t\in \mathcal{T}^{p}} F^{\alpha}_{p,q}(t)<\infty\;. \end{align}\] Since \((F^{\alpha}_{p,q}(t_{n}))_{n\in\mathbb{N}}\) is a bounded sequence, for sufficiently large \(n\geq N\in\mathbb{N}\) each gradient \(\nabla^{g}t_{n}\) is timelike almost everywhere in \(\Omega\). Moreover, coercivity of \(F^{\alpha}_{p,q}\) implies that the sequence \((t_{n})_{n\in\mathbb{N}}\) is bounded in \(H^{p}(\Omega)\). By the sequential Banach-Alaouglu Theorem (\(H^{p}(\Omega)\) is a reflexive Banach space) and weak closedness of \(\mathcal{T}^{p}\) (cf. Remark 5), there exists a subsequence \((t_{n_{k}})_{n_{k}\in\mathbb{N}}\) in \(\mathcal{T}^{p}\) which converges weakly to a function \(t_{\alpha}\in \mathcal{T}^{p}\). However, even if \(t_{n_{k}}\in \mathcal{T}^{p,t}\) for each \(n_{k}\geq N\), a priori, \(\nabla^{g}t_{\alpha}\) could still be null or vanish a.e. on \(\Omega\). Nevertheless, the assumption that \(p\geq2\) rules out this possibility.

Indeed, as in the proof of the previous lemma, by the Rellich-Kondrachov theorem the bounded and weakly convergent subsequence \((t_{n_{k}})_{n_{k}\in\mathbb{N}}\) converges (after passing to a further subsequence) strongly in \(H^{1}(\Omega)\) to \(t_{\alpha}\) and also \(|\nabla^{g}t_{n_{k}}|^{2}_{g}\) converges to \(|\nabla^{g}t_{\alpha}|^{2}_{g}\) a.e. in \(\Omega\). We now show that this actually implies that \(\nabla^{g}t_{\alpha}\) must be timelike a.e. Seeking a contradiction, assume that the subset \(A:=\{x\in\Omega:|\nabla^{g}t_{\alpha}|_{g}^{2}(x)=0\}\) is not a null set. Then, \((|\nabla^{g}t_{n_{k}}|^{-2q}_{g})_{k\in\mathbb{N}}\) is a sequence of measurable non-negative functions which diverge to \(+\infty\) a.e. in \(A\). Using Fatou’s lemma \[\begin{align} \liminf_{k\to\infty} F_{g,q}(t_{n_{k}})&\geq\liminf_{k\to\infty} \int_{A}\frac{1}{|\nabla^{g}t_{n_{k}}|_{g}^{2q}}d\mu_{h}\geq\int_{A}\liminf_{k\to\infty}\frac{1}{|\nabla^{g}t_{n_{k}}|_{g}^{2q}}d\mu_{h}=+\infty\;, \end{align}\] which contradicts that \(\lim_{k\to\infty}F^{\alpha}_{p,q}(t_{n_{k}})=F_{\textrm{inf}}<\infty\) since \(F_{g,q}(t_{n_{k}})\leq \alpha^{-1}F^{\alpha}_{p,q}(t_{n_{k}})\). Hence, \(t_{\alpha}\in\mathcal{T}^{p,t}\).

Since \(t_{n},t_{\alpha}\in \mathcal{T}^{p,t}\) for \(n\geq N\) and \(F^{\alpha}_{p,q}\) is weakly lower semicontinuous on \(\mathcal{T}^{p,t}\), it follows that \(t_{\alpha}\) is a minimizer of the functional (i.e. \(F_{\inf}=F^{\alpha}_{p,q}(t_{\alpha})\)), \[\begin{align} F_{\inf}\leq F^{\alpha}_{p,q}(t_{\alpha})\leq \liminf_{k\to\infty} F^{\alpha}_{p,q}(t_{n_{k}})=F_{\inf}\;. \end{align}\] Uniqueness of the minimizer then follows by strict convexity of \(F^{\alpha}_{p,q}\) on \(\mathcal{T}^{p,t}\).

In the case that \(\alpha=0\), existence and uniqueness of a minimizer \(t_{0}\) of \(F^{\alpha}_{p,q}\) is a direct application of the direct method in the calculus of variations using that \(F_{h,p}\) is coercive, strictly convex and weakly lower semicontinuous on the set \(\mathcal{T}^{p}\) (cf. Lemma 8). In this case the arguments apply already for \(p\geq1\). However, the gradient of the minimizer can be null or even vanish. ◻

Proof. For \(k\in\mathbb{N}\), \(\gamma\in [0,1)\) and \(p>\frac{n}{2}+k+\gamma\), it follows by the Sobolev embedding theorem that \(t\in C^{k,\gamma}(\Omega)\). Assume that \(\nabla^{g}t\) is null or vanishes at \(x_{0}\in \Omega\) and consider a chart \((\varphi,U)\) around this point. We define the \(C^{k-1,\gamma}\) regular function \[\begin{align} \label{eq:definition95f} f:= (-g(\nabla^{g}t,\nabla^{g}t))\circ\varphi^{-1} : \varphi(U')\subset \mathbb{R}^{n} \rightarrow [0,\infty)\;, \end{align}\tag{5}\] where \(U':=U\cap \Omega\), \(f\) is non-negative because \(\nabla^{g}t\) is a.e. timelike in \(\Omega\) (and everywhere causal by continuity of \(\nabla^{g}t\)) and without loss of generality \(y_{0}:=\varphi(x_{0})=0\), so \(f(0)=0\).

Since the boundary \(\partial\Omega\) is Lipschitz continuous (so, locally, \(\Omega\) lies on one side of \(\partial\Omega\)), for any \(x_{0}\in \Omega\) there exists a positive measure set \(\Gamma\subseteq \mathbb{S}^{n-1}\) such that, for a sufficiently small \(\lambda>0\), \((0,\lambda)\times \Gamma\subset \varphi(U')\). Then, for spherical coordinates \((r,\omega)\in (0,\lambda)\times\Gamma\) \[\begin{align} \label{eq:bound95fbeta} f(r,\omega)\leq C r^{\beta} \implies \frac{1}{f(r,\omega)}\geq \frac{1}{Cr^{\beta}}\;\textrm{a.e. in}\; (0,\lambda)\times \Gamma\;, \end{align}\tag{6}\] where the inequality on the right-hand side is well-defined a.e. since \(f\) can vanish at most on a null set. Then 6 gives \[\begin{align} &\int_{\varphi(U')}\frac{1}{(f(y))^{q}}d\mu_{h}\geq C\int_{(0,\lambda)\times \Gamma}\frac{1}{(f(y))^{q}}d^{n}y\geq C\int_{0}^{\lambda}\frac{r^{n-1}}{r^{\beta q}}dr \end{align}\] where we used that \(d\mu_{h}=\rho(y) d^{n}y\geq C d^{n}y\) in \(\varphi(U')\) with \(d^{n}y\) the Lebesgue measure on \(\mathbb{R}^{n}\), \(\rho\) positive and smooth and \(C>0\) (note that in this proof constants change but are denoted with the same symbol). In particular, the above expression diverges if the exponent is smaller or equal to \(-1\), i.e. whenever \[\begin{align} q\geq n/\beta\;. \end{align}\] Expression 6 is satisfied in the different mentioned cases:

• If \(p>n/2+1+\gamma\) with \(\gamma\in(0,1)\), then \(f\in C^{0,\gamma}(\varphi(U'))\). Hence, \[\begin{align} \label{eq:bound95f0} f(y)\leq C\|y\|^{\gamma}= Cr^{\gamma},\quad\textrm{with}\quad C\in(0,\infty)\;, \end{align}\tag{7}\] i.e. \(f\) satisfies 6 with \(\beta=\gamma\) and the integral diverges for \(q\geq n/\gamma\).

• If \(p> n/2+2\), then \(f\) is in particular locally Lipschitz continuous and expression 7 with \(\gamma=1\) holds. Hence, in this case the integral diverges³ for \(q\geq n\).

• Let \(p>n/2+3\), \(\nabla^{g}t\) be past-directed causal in an open subset \(\Omega'\), with \(\Omega\subset\Omega'\) and \(U'\) a neighbourhood around \(x_{0}\) with \(U'\subset \Omega'\) and consider again the function \(f\) given by 5 . Then, \(y_{0}:=\varphi(x_{0})\) is a local minimum and \(Df(y_{0})=0\), since \(f\) is non-negative in \(\varphi(U')\) (\(\nabla^{g}t\) is causal in \(\Omega'\)), also if \(x_{0}\in\partial\Omega\). The Taylor expansion of \(f\) around \(y_{0}=0\) yields \[\begin{align} f(y)&=\sum_{|\alpha|=2}\frac{1}{\alpha!}\frac{\partial^{\alpha}f}{\partial y^{\alpha}}(0)y^{\alpha}+R(y),\quad \textrm{with}\quad R(y)=o(\|y\|^{2}) \quad \textrm{as} \;y\to0\;. \end{align}\] In spherical coordinates, for sufficiently small \(\lambda>0\) and \(\Gamma\subset \mathbb{S}^{n-1}\), \[\begin{align} f(r,\omega) \leq Cr^{2}, \quad (r,\omega)\in(0,\lambda)\times \Gamma\;. \end{align}\] Hence, \(\beta=2\) and the integral diverges for \(q\geq n/2\).

 ◻

Proof. In order to determine the Euler-Lagrange equation, one has to compute the Gateaux derivative of \(F^{\alpha}_{p,q}\) in those directions \(\eta\) for which \(t_{\alpha}+s\eta\) is in the set \(\mathcal{T}^{p}\) for sufficiently small \(|s|\in[0,\infty)\). In particular, this holds for any \(\eta\in H^{p}(\Omega)\supset \mathcal{T}^{p}\) which satisfies the zero-mean condition since the gradient of the minimizer \(t_{\alpha}\) is everywhere timelike and thus \(t_{\alpha}\) is an interior point of \(\mathcal{T}^{p}\). Then, for any such direction \(\eta\) the following variational equality holds \[\begin{align} \label{eq:variational95eq} DF^{\alpha}_{p,q}(t_{\alpha})[\eta]=0\;. \end{align}\tag{8}\] However, in order to view \(t_{\alpha}\) as the weak interior solution to a partial differential equation, it is convenient to actually consider arbitrary test functions \(\varphi\in C^{\infty}_{0}(\Omega^{\circ})\) (with \(\Omega^{\circ}\) the topological interior of \(\Omega\)). This is possible, since for any such \(\varphi\), the function \(\eta_{\varphi}\) defined by \[\begin{align} \eta_{\varphi}:=\varphi-\frac{1}{\mu_{h}(\Omega)}\int_{\Omega}\varphi \;d\mu_{h}\;, \end{align}\] satisfies the zero-mean condition and, as \(\nabla^{g}\varphi=\nabla^{g}\eta_{\varphi}\), equation 8 also holds for any \(\varphi\in C^{\infty}_{0}(\Omega^{\circ})\). Thus we consider such test functions when computing the Gateaux derivatives.

The Gateaux derivative of \(F_{h,p}\) in the direction of \(\varphi\in C^{\infty}_{0}(\Omega^{\circ})\) is \[\begin{align} \label{eq:Gateaux95diff95Fhp} DF_{h,p}(t_{\alpha})[\varphi]&=\frac{d}{ds}\Big(\sum_{i=0}^{p-1}\int_{\Omega}h\big((\nabla^{h})^{i}(u-\nabla^{g} (t_{\alpha}+s\varphi),(\nabla^{h})^{i}(u-\nabla^{g} (t_{\alpha}+s\varphi)\big)\big)\Big|_{s=0}d\mu_{h} \nonumber\\ &=-2\sum_{i=0}^{p-1}\int_{\Omega} h((\nabla^{h})^{i}\nabla^{g}\varphi,(\nabla^{h})^{i}(u-\nabla^{g} t_{\alpha}))d\mu_{h}\;, \end{align}\tag{9}\] which, integrating by parts, can also be rewritten in the distributional sense as \[\begin{align} \label{eq:weak95EL} DF_{h,p}(t_{\alpha})[\varphi]=-2\sum_{i=0}^{p-1}\int_{\Omega} \varphi\;\nabla^{g\ast}(\nabla^{h\ast})^{i}(\nabla^{h})^{i}(u-\nabla^{g} t_{\alpha})d\mu_{h}\;, \end{align}\tag{10}\] where \(\nabla^{g\ast}\) and \(\nabla^{h\ast}\) denote the adjoints of \(\nabla^{g}\) and \(\nabla^{h}\) with respect to the Riemannian \(h\)-scalar product (cf. [31]). We now proceed analogously for \(F_{g,q}\), \[\begin{align} \label{eq:Lorentzian95functional} DF_{g,q}(t_{\alpha})[\varphi]&=2q\int_{\Omega}\frac{g(\nabla^{g} t_{\alpha},\nabla^{g} \varphi)}{|\nabla^{g} t_{\alpha}|_{g}^{2q+2}}d\mu_{h}=2q\int_{\Omega}\frac{h(\nabla^{g}t_{\alpha},\nabla^{h}\varphi)}{|\nabla^{g} t_{\alpha}|_{g}^{2q+2}}d\mu_{h}\nonumber\\ &=-2q\int_{\Omega}\varphi\;{\rm{div}}_{h}\Big(\frac{\nabla^{g} t_{\alpha}}{|\nabla^{g} t_{\alpha}|_{g}^{2q+2}}\Big) d\mu_{h}\;, \end{align}\tag{11}\] where in the second step we used that \(g(\nabla^{g}\varphi,X)=X(\varphi)=h(\nabla^{h}\varphi,X)\), with \(X:=\nabla^{g}t_{\alpha}\in\Gamma(\Omega)\), and \({\rm{div}}_{h}(Y)=-(\nabla^{h})^{\ast}Y\) for \(Y\in \Gamma(\Omega)\). Moreover, \(|\nabla^{g}t_{\alpha}|_{g}^{-2q-2}\) is integrable because \(\nabla^{g}t_{\alpha}\) is everywhere timelike and, since \(\Omega\) is compact, the norm \(|\nabla^{g}t_{\alpha}|^{2}_{g}\) can be bounded away from zero. Hence, for any \(\varphi\in C^{\infty}_{0}(\Omega^{\circ})\) the following weak Euler-Lagrange equation holds: \[\begin{align} \int_{\Omega} \varphi\Big(\sum_{i=0}^{p-1}\nabla^{g\ast}(\nabla^{h\ast})^{i}(\nabla^{h})^{i}(\nabla^{g} t_{\alpha}-u)-\alpha q\;{\rm{div}}_{h}\Big(\frac{\nabla^{g} t_{\alpha}}{|\nabla^{g} t_{\alpha}|_{g}^{2q+2}}\Big)\Big) d\mu_{h}=0\;. \end{align}\] I.e. \(t_{\alpha}\) is a weak solution to the following quasi-linear partial differential equation \[\begin{align} \label{eq:PDE} \sum_{i=0}^{p-1}L_{h,i}t_{\alpha}-\alpha qL_{g}t_{\alpha}=f(u)\;, \end{align}\tag{12}\] in \(\Omega^{\circ}\), where \(L_{h,i}\) and \(L_{g}\) are differential operators acting on the minimizer \(t_{\alpha}\) and \(f(u)\) a smooth function depending on the vector field \(u\) \[\begin{align} &\begin{cases} L_{h,i}t:=D_{i}^{\ast}D_{i}t\\ L_{g}t:={\rm{div}}_{h}(|\nabla^{g}t|_{g}^{-2q-2}\nabla^{g}t) \end{cases} &f(u):=\sum_{i=0}^{p-1}\nabla^{g\ast}(\nabla^{h\ast})^{i}(\nabla^{h})^{i}u \end{align}\] with \(D_{i}:=(\nabla^{h})^{i}\nabla^{g}\). The above PDE has order \(2p\) and \(L_{h,p-1}=D^{\ast}_{p-1}D_{p-1}\) is its higher order term (\(L_{g}\) is only of second order). In local coordinates, \(L_{h,p-1}\) is \[\begin{align} \label{eq:higher95order95operator} &(D_{p-1}t)^{i}_{j_{1}\ldots j_{p-1}}=g^{ik}\partial_{j_{1}}\ldots \partial_{j_{p-1}}\partial_{k}t +\textrm{(lower order terms)}\nonumber \\ &L_{h,p-1}t\!=\!(-1)^{p}g^{lk}g^{mi}h_{li}h^{i_{1}j_{1}}\ldots h^{i_{p-1}j_{p-1}} \partial_{k}\partial_{m}\partial_{i_{1}}\ldots \partial_{i_{p-1}}\partial_{j_{1}} \ldots \partial_{j_{p-1}}t +\textrm{(l.o.t)} \end{align}\tag{13}\] where the adjoint \(D_{p-1}^{\ast}\) was computed using local coordinates and integrating by parts \(p\)-times against a function \(\varphi\in C^{\infty}_{0}(\Omega)\): \[\begin{align} &\int_{\Omega}\varphi\;L_{h,p-1}t\;d\mu_{h}=\int_{\Omega}\varphi\;D_{p-1}^{\ast}D_{p-1}t\;d\mu_{h}=\int_{\Omega}h(D_{p-1}t,D_{p-1}\varphi)\;d\mu_{h}\\ &=\int_{\Omega} h_{li}h^{i_{1}j_{1}}\ldots h^{i_{p-1}j_{p-1}} (D_{p-1}t)^{l}_{i_{1} \ldots i_{p-1}}g^{ik}\partial_{j_{1}} \ldots \partial_{j_{p-1}}\partial_{k}\varphi\;d\mu_{h} +\textrm{(l.o.t.)}\\ &=(-1)^{p}\int_{\Omega} \varphi\;h_{li}g^{ik}h^{i_{1}j_{1}}\ldots h^{i_{p-1}j_{p-1}} \partial_{j_{1}} \ldots \partial_{j_{p-1}}\partial_{k}(D_{p-1}t)^{l}_{i_{1} \ldots i_{p-1}}\;d\mu_{h} +\textrm{(l.o.t.)}\;. \end{align}\] From expression 13 the principal symbol (using the conventions in [32]) for the Euler-Lagrange equation satisfies that \[\begin{align} (-1)^{p}\sigma_{2p}(L_{h,p-1})(\xi)&=g^{lk}g^{mi}h_{li}h^{i_{1}j_{1}}\ldots h^{i_{p-1}j_{p-1}} \xi_{k}\xi_{m}\xi_{i_{1}}\ldots \xi_{i_{p-1}}\xi_{j_{1}} \ldots \xi_{j_{p-1}}\\ &=|\xi|_{h}^{2p-2}h(\xi^{\sharp,g},\xi^{\sharp,g})\geq C|\xi|_{h}^{2p}\;. \end{align}\] with \(\xi\in T_{x}^{\ast}\Omega\), \((\xi^{\sharp,g})^{i}:=g^{ij}\xi_{j}\) and, as \(h_{x}(\xi^{\sharp,g}(x),\xi^{\sharp,g}(x))=(g^{lk}g^{mi}h_{li}\xi_{k}\xi_{m})(x)\) is positive definite on \(T^{\ast}_{x}\Omega\) (for each \(x\in \Omega^{\circ}\)) there exists a constant \(C(x)>0\) such that \(h(\xi^{\sharp,g},\xi^{\sharp,g})(x)\geq C(x)|\xi|_{h}^{2}(x)\) and which can be chosen independent of the base-point \(x\in \Omega\) by compactness of \(\Omega\). Thus, the Euler-Lagrange equation is uniformly elliptic. In order to apply interior regularity theory, we rewrite the elliptic equation as \[\begin{align} Lt_{\alpha}:=\sum_{i=0}^{p-1} L_{h,i}t_{\alpha}=\tilde{f}\;, \end{align}\] where \(\tilde{f}:=f(u)+\alpha qL_{g}t_{\alpha}\). In particular, since \(\nabla^{g}t_{\alpha}\) is everywhere timelike in the compact set \(\Omega\) and \(p-1>n/2\) (so the Sobolev multiplication theorems apply), it follows that \(|\nabla^{g}t_{\alpha}|_{g}^{-2q-2}\nabla^{g}t_{\alpha}\in H^{p-1}(\Omega)\) and so \(\tilde{f}\in H^{p-2}(\Omega)\). Since \(L\) is a uniformly elliptic operator of order \(2p\) and \(Lt_{\alpha}\in H^{p-2}(\Omega)\), the \(L^{2}\) based interior elliptic regularity theory implies that \(t_{\alpha}\in H_{\textrm{loc}}^{3p-2}(\Omega)\) (cf. [33]). An iteration of this argument and the Sobolev embedding theorems yield interior smoothness of the minimizer \(t_{\alpha}\) because, for arbitrary \(r\in\mathbb{N}\), if \(Lt_{\alpha}\in H^{r-2}_{\textrm{loc}}(\Omega)\), then \(t_{\alpha}\in H_{\textrm{loc}}^{2p+r-2}(\Omega)\). ◻

Proof. Let \(p\geq2\). Since \(t_{p,q}\) is a minimizer of \(F^{\alpha}_{p,q}\) and \((M,g)\) is stably causal there exists a temporal function \(\tau\) with (possibly after a reparametrization) \(|\nabla^{g}\tau|_{g}\geq 1\) and a \(q\)-independent constant \(C\in(0,\infty)\) such that \[\begin{align} \label{eq:uniform95bound} F^{\alpha}_{p,q}(t_{p,q})\leq F^{\alpha}_{p,q}(\tau)\leq C\;. \end{align}\tag{14}\] Boundedness of the real-valued sequence \((F^{\alpha}_{p,q}(t_{p,q}))_{q\in\mathbb{N}}\) implies, by coercivity of the misalignment functional, that the sequence \((t_{p,q})_{q\in\mathbb{N}}\) is bounded in the \(H^{p}(\Omega)\)-norm. By the Banach-Alaoglu theorem and weak closedness of \(\mathcal{T}^{p}\), there exists a subsequence \((t_{p,q_{k}})_{k\in\mathbb{N}}\) converging weakly to a limit function \(t_{p,\infty}\in\mathcal{T}^{p}\). By the Rellich-Kondrachov theorem (after passing to a further subsequence) \((t_{p,q_{k}})_{k\in\mathbb{N}}\) converges also strongly in \(H^{1}(\Omega)\), which implies a.e. convergence of (a subsequence of) \(|\nabla^{g}t_{p,q_{k}}|_{g}\) to \(|\nabla^{g}t_{p,\infty}|_{g}\).
In order to prove expression ?? , consider the following functional \[\begin{align} \label{eq:ess95inf95def} E(t):=\underset{x\in\Omega}{\text{\rm{ess inf}}}\;|\nabla^{g}t|_{g}(x)\;, \end{align}\tag{15}\] with \(t\in\mathcal{T}^{p}\). Let \(\varepsilon>0\). By the definition of the essential infimum the following set \[\begin{align} A_{\varepsilon}:=\{x\in\Omega:|\nabla^{g}t_{p,\infty}|_{g}(x)\leq E(t_{p,\infty})+\varepsilon\} \end{align}\] has positive measure. Since \(|\nabla^{g}t_{p,q_{k}}|_{g}\to|\nabla^{g}t_{p,\infty}|_{g}\) a.e. in \(A_{\varepsilon}\), by the Egorov theorem, there exists a positive measure subset \(B_{\varepsilon}\subset A_{\varepsilon}\) in which the convergence is uniform. Thus, using the triangle inequality (for the absolute value) there exists a \(K\in \mathbb{N}\) such that for all \(x\in B_{\varepsilon}\) and \(k\geq K\) \[\begin{align} \label{eq:essinf95estimate} |\nabla^{g}t_{p,q_{k}}|_{g}(x)&\leq\big|\;|\nabla^{g}t_{p,q_{k}}|_{g}(x)-|\nabla^{g}t_{p,\infty}|_{g}(x)\big|+|\nabla^{g}t_{p,\infty}|_{g}(x)\leq E(t_{p,\infty})+2\varepsilon \end{align}\tag{16}\] Then, since \(\mu_{h}(B_{\varepsilon})>0\) for \(k\geq K\) it holds that \[\begin{align} E(t_{p,q_{k}})\leq E(t_{p,\infty})+2\varepsilon \implies \limsup_{k\to\infty} E(t_{p,q_{k}})\leq E(t_{p,\infty})+2\varepsilon\;, \end{align}\] taking the \(\varepsilon\to0\) limit gives back expression ?? for the subsequence \((t_{p,q_{k}})_{q_{k}\in\mathbb{N}}\). Once convergence of \((t_{p,q})_{q\in\mathbb{N}}\) is proven, the estimate holds for the full sequence.

Secondly, we prove bound ?? . Assume that \(E(t_{p,\infty})<1\) and choose a \(\varepsilon>0\) small enough such that \(E(t_{p,\infty})+2\varepsilon<1\). From expression 16 it then follows that \[\begin{align} \label{eq:limsup952} \!\!\!F_{g,q_{k}}(t_{p,q_{k}})=\int_{\Omega}\frac{d\mu_{h}}{|\nabla^{g}t_{p,q_{k}}|_{g}^{2q_{k}}}\! \geq\! \int_{B_{\varepsilon}}\frac{1}{|\nabla^{g}t_{p,q_{k}}|_{g}^{2q_{k}}}d\mu_{h}\geq \mu_{h}(B_{\varepsilon}) (E(t_{p,\infty})+2\varepsilon)^{-2q_{k}}\! , \end{align}\tag{17}\] which diverges as \(k\to \infty\), contradicting 14 .

In the third place, we show convergence of the sequence of minimizers \((t_{p,q})_{q\in\mathbb{N}}\). Note that \(\mathcal{S}^{p}\) is a convex set because of the reversed triangle inequality. Let \(t,t'\in \mathcal{S}^{p}\) be arbitrary and \(\lambda\in(0,1)\). Then, \(t_{\lambda}:=\lambda t'+(1-\lambda)t\in \mathcal{S}^{p}\) since \[\begin{align} &|\nabla^{g}t_{\lambda}|_{g}\geq \lambda|\nabla^{g}t'|_{g}+(1-\lambda)|\nabla^{g}t|_{g}\implies E(t_{\lambda})\geq \lambda E(t')+(1-\lambda)E(t)\geq1\;. \end{align}\] In particular, consider the case with \(|\nabla^{g}t'|_{g}>1\), for which \(E(t_{\lambda})>1\). Then, \(F_{g,q_{k}}(t_{\lambda})\to0\) since \(F_{g,q_{k}}(t_{\lambda})\leq \mu_{h}(\Omega)(E(t_{\lambda}))^{-2q_{k}}\to0\) as \(k\to\infty\)). Using that \(t_{p,q}\) minimizes \(F^{\alpha}_{p,q}\) it follows that: \[\begin{align} \label{eq:limsup95Fhp} &\limsup_{k\to\infty}F_{h,p}(t_{p,q_{k}})\leq \limsup_{k\to\infty}(F_{h,p}(t_{\lambda})+\alpha F_{g,q_{k}}(t_{\lambda}))= F_{h,p}(t_{\lambda})\nonumber\\ &\implies \limsup_{k\to\infty}F_{h,p}(t_{p,q_{k}})\leq F_{h,p}(t)\;, \end{align}\tag{18}\] where we used that, \(t_{\lambda}\to t\) as \(\lambda\to0\). Using weak lower semicontinuity of \(F_{h,p}\) yields \[\begin{align} F_{h,p}(t_{p,\infty})\leq \liminf_{k\to\infty} F_{h,p}(t_{p,q_{k}})\leq\limsup_{k\to\infty} F_{h,p}(t_{p,q_{k}})\leq F_{h,p}(t)\;. \end{align}\] Since \(t\in \mathcal{S}^{p}\) was arbitrary and \(t_{p,\infty}\in \mathcal{S}^{p}\), we conclude that \[\begin{align} F_{h,p}(t_{p,\infty})=\min_{t\in \mathcal{S}^{p}} F_{h,p}(t)\;, \end{align}\] i.e. \(t_{p,\infty}\) is the minimizer of \(F_{h,p}\) on the set \(\mathcal{S}^{p}\). By strict convexity of \(F_{h,p}\), the minimizer is unique. Hence, any subsequence of the \(H^{p}(\Omega)\)-bounded sequence \((t_{p,q})_{q\in\mathbb{N}}\) admits a further subsequence converging weakly in \(H^{p}(\Omega)\) and strongly in \(H^{1}(\Omega)\) (by Banach-Alaoglu and Rellich-Kondrachov) to the unique minimizer \(t_{p,\infty}\) of \(F_{h,p}\) on \(\mathcal{S}^{p}\). This implies convergence of the full sequence \((t_{p,q})_{q\in\mathbb{N}}\).

Finally, if \(p>n/2+1\) then the bounded (in \(H^{p}(\Omega)\)) sequence \((t_{p,q})_{q\in\mathbb{N}}\) has a subsequence \((t_{p,q_{k}})_{q_{k}\in\mathbb{N}}\) converging in \(C^{1}(\Omega)\) to \(t_{p,\infty}\in C^{1}(\Omega)\) (that the limit has to be \(t_{p,\infty}\) follows from convergence of \((t_{p,q})_{q\in\mathbb{N}}\) and uniqueness of limits). Since this holds for any subsequence, the full sequence \((t_{p,q})_{q\in\mathbb{N}}\) converges in \(C^{1}\), so the essential infimums in ?? become infimums. Then, \(C^{0}\)-convergence of \(|\nabla^{g}t_{p,q}|_{g}\) and the bound \[\begin{align} \big| \inf_{x\in\Omega}|\nabla^{g}t_{p,q}|_{g}-\inf_{x\in\Omega}|\nabla^{g}t_{p,\infty}|_{g}\big|\leq \||\nabla^{g}t_{p,q}|_{g}-|\nabla^{g}t_{p,\infty}|_{g}\|_{C^{0}(\Omega)}\;, \end{align}\] imply convergence of the sequence \(\big(\inf_{x\in\Omega}|\nabla^{g}t_{p,q}|_{g}\big)_{q\in\mathbb{N}}\) to the infimum of \(\nabla^{g}t_{p,\infty}\). ◻

Proof. Consider again expression 18 with \(t=t_{p,\infty}\in\mathcal{S}^{p}\). Using weak lower semicontinuity of \(F_{h,p}\) it follows that \[\begin{align} \limsup_{q\to\infty}F_{h,p}(t_{p,q})\leq F_{h,p}(t_{p,\infty})\leq \liminf_{q\to\infty}F_{h,p}(t_{p,q})\;, \end{align}\] i.e. \(F_{h,p}(t_{p,q})\to F_{h,p}(t_{p,\infty})\) as \(q\to\infty\). Hence, the sequence \((\|X_{q}\|_{H^{p-1}(\Omega,h)})_{q\in\mathbb{N}}\) converges to \(\|X\|_{H^{p-1}(\Omega,h)}\), with \(X_{q}:=u-\nabla^{g}t_{p,q}\) and \(X_{\infty}:=u-\nabla^{g}t_{p,\infty}\), and by the previous proposition \(X_{q}\) also converges weakly in \(H^{p-1}\) to \(X_{\infty}\). Since \(H^{p-1}(\Omega)\) is a scalar product space, \(X_{q}\) converges also strongly in \(H^{p-1}\) to \(X_{\infty}\) as follows from a direct inspection of the following expression \[\begin{align} \|X_{q}-X_{\infty}\|_{H^{p-1}(\Omega)}^{2}=\|X_{q}\|_{H^{p-1}(\Omega)}^{2}+ \| X_{\infty}\|^{2}_{H^{p-1}(\Omega)}-2(X_{q},X_{\infty})_{H^{p-1}(\Omega)}\;, \end{align}\] where the right hand side converges to \(0\) as \(q\to\infty\) by convergence of the norms and weak convergence in \(H^{p-1}(\Omega)\). Strong \(H^{p-1}(\Omega)\) convergence of the sequence of gradients together with strong \(H^{1}(\Omega)\) convergence of the sequence of functions, allows us to upgrade the latter to strong \(H^{p}(\Omega)\) convergence.
For the second claim, let \(\varphi\in C^{\infty}_{0}(\Omega^{\circ})\) and denote its support by \(K\). By the proof of Proposition 12 it is not necessary that they satisfy the zero-mean condition because the functional only depends on weak derivatives of \(t\in\mathcal{S}^{p}\). Then, there exists a sufficiently small \(s'\in [0,1]\) such that \(t_{p,\infty}+s\varphi\in \mathcal{S}^{p}\) for all \(|s|\in[0,s']\) (obvious in \(\Omega\setminus K\)). Indeed, condition ?? implies existence of a \(\delta>0\) such that \(|\nabla^{g}t_{p,\infty}|_{g}(x)\geq 1+\delta\) for all \(x\in\Omega\) and so in \(K\) it holds that \[\begin{align} |\nabla^{g}t_{p,\infty}+s\nabla^{g}\varphi|_{g}^{2}&=|\nabla^{g}t_{p,\infty}|_{g}^{2}-2sg(\nabla^{g}t_{p,\infty},\nabla^{g}\varphi)-s^{2}g(\nabla^{g}\varphi,\nabla^{g}\varphi)\\ &\geq|\nabla^{g}t_{p,\infty}|_{g}^{2}-2|s| |g(\nabla^{g}t_{p,\infty},\nabla^{g}\varphi)|-s^{2}|g(\nabla^{g}\varphi,\nabla^{g}\varphi)|\\ &\geq(1+\delta)^{2}-|s|C(2|\nabla^{g}t_{p,\infty}|_{h}|\nabla^{g}\varphi|_{h}+|s||\nabla^{g}\varphi|_{h}^{2})\\ &\geq(1+\delta)^{2}-|s|C(C')^{2}(2+|s|)\;, \end{align}\] where in the third line we simply used that since \(g\) is smooth and the support of \(\varphi\) is compact there exists a constant \(C>0\) such that \(|g(\nabla^{g}t_{p,\infty},\nabla^{g}\varphi)|\leq C|\nabla^{g}t_{p,\infty}|_{h}|\nabla^{g}\varphi|_{h}\) in \(K\) and in the fourth line we exploited again compactness of \(K\) and continuity of \(\nabla^{g}t_{p,\infty}\) and \(\nabla^{g}\varphi\) to upper bound their norms by a constant \(C'\). Hence, one can now choose an \(s'\in[0,1]\) sufficiently small such that \[\begin{align} \label{eq:s95estimate} s'C(C')^{2}(2+s')< (1+\delta)^{2}-1\;, \end{align}\tag{19}\] and then \(|\nabla^{g}t_{p,\infty}+s'\nabla^{g}\varphi|_{g}^{2}>1\) in \(\Omega\), so \(t_{p,\infty}+s'\varphi\in \mathcal{S}^{p}\). Note that the function \(f(s'):=s'(2+s')\) is strictly increasing for \(s'\geq0\) (\(f'(s')=2+2s'>0\)) so 19 is satisfied by any \(|s|\in[0,s']\) and thus \(t_{p,\infty}+s\varphi\in \mathcal{S}^{p}\) for all \(s\in[-s',s']\). Then, the Gateaux derivative of \(F_{h,p}\) at \(t_{p,\infty}\) in the direction of \(\varphi\) vanishes and, using the proof of Proposition 12, the following weak Euler-Lagrange equation is satisfied on \(\Omega^{\circ}\) \[\begin{align} \sum_{i=0}^{p-1}\nabla^{g\ast}(\nabla^{h\ast})^{i}(\nabla^{h})^{i}(\nabla^{g} t_{p,\infty}-u)=0\;, \end{align}\] which is again uniformly elliptic, yielding smoothness of \(t_{p,\infty}\) in the interior of \(\Omega\). ◻

Proof. For the first claim note that if \(\tilde{t}_{p}\) is almost everywhere steep, then \(\tilde{t}_{p}\in \mathcal{S}^{p}\) and since it minimizes \(F_{h,p}\) over \(\mathcal{T}^{p}\) it also minimizes the functional over \(\mathcal{S}^{p}\). Uniqueness of the minimizer and Proposition 15 then imply that \(\tilde{t}_{p}=t_{p,\infty}\).

Assume that \(E(\tilde{t}_{p})<1\) and \(E(t_{p,\infty})>1\). Since \(\mathcal{T}^{p}\) is convex, any convex combination of \(\tilde{t}_{p}\) and \(t_{p,\infty}\) belongs again to \(\mathcal{T}^{p}\). Applying the reversed triangle inequality to \(t_{\lambda}:=\lambda \tilde{t}_{p}+(1-\lambda)t_{p,\infty}\), with \(\lambda\in[0,1]\), leads to \[\begin{align} E(t_{\lambda})\geq \lambda E(\tilde{t}_{p})+(1-\lambda)E(t_{p,\infty})\to E(t_{p,\infty})>1 \quad \textrm{as}\quad\lambda\to0^{+}\;. \end{align}\] Hence, there exists a \(\lambda'\in[0,1]\) such that \(E(t_{\lambda})\geq 1\) for all \(\lambda\in[0,\lambda']\), which implies that \(t_{\lambda}\in \mathcal{S}^{p}\) for \(\lambda\in [0,\lambda']\). On the other hand, using strict convexity of \(F_{h,p}\) we have that \[\begin{align} F_{h,p}(t_{\lambda})<\lambda F_{h,p}(\tilde{t}_{p})+(1-\lambda)F_{h,p}(t_{p,\infty})< F_{h,p}(t_{p,\infty})\;, \end{align}\] where we used that \(\tilde{t}_{p}\) is the minimizer of \(F_{h,p}\) over \(\mathcal{T}^{p}\) and \(\tilde{t}_{p}\neq t_{p,\infty}\). However, for \(\lambda\in[0,\lambda']\), \(t_{\lambda}\in \mathcal{S}^{p}\) so the above expression contradicts that \(t_{p,\infty}\) is the minimizer of \(F_{h,p}\) over \(\mathcal{S}^{p}\). Therefore, \(E(t_{p,\infty})=1\).

Finally, if \(u=\nabla^{g}f\) and \(f\) satisfies the zero-mean condition (if not, one can subtract a constant such that it does satisfy it), then \(f\) is the unique minimizer of \(F_{h,p}\) on \(\mathcal{T}^{p}\) by strict convexity of the functional. Then, part (i) and (ii) imply that \[\begin{align} E(t_{p,\infty})\geq \inf_{x\in\Omega}|u|_{g}(x)\;. \end{align}\] ◻

Proof. For \(s\in(0,1)\), \(t_{\alpha}+s(\tilde{t}_{\alpha}-t_{\alpha})\in\mathcal{T}^{p}\) and \(\tilde{t}_{\alpha}+s(t_{\alpha}-\tilde{t}_{\alpha})\in\mathcal{T}^{p}\) by convexity of \(\mathcal{T}^{p}\). Since \(p> n/2+1+\gamma\) and \(q\geq n/\gamma\), the gradient of the minimizers is everywhere timelike and the functional \(F_{g,q}\) is Gateaux differentiable. In particular, convexity of \(F_{g,q}\) implies that \[\begin{align} &F_{g,q}(t_{\alpha})\geq F_{g,q}(\tilde{t}_{\alpha})+DF_{g,q}(\tilde{t}_{\alpha})[t_{\alpha}-\tilde{t}_{\alpha}]\implies DF_{g,q}(\tilde{t}_{\alpha})[t_{\alpha}-\tilde{t}_{\alpha}] + DF_{g,q}(t_{\alpha})[\tilde{t}_{\alpha}-t_{\alpha}] \leq0 \end{align}\] Let \(\tilde{F}_{h,p}(t):=\|\tilde{u}-\nabla^{g}t\|^{2}_{H^{p-1}(\Omega)}\) and recall that \(F_{h,p}(t):=\|u-\nabla^{g}t\|^{2}_{H^{p-1}(\Omega)}\) . Then, since \(t_{\alpha}\) and \(\tilde{t}_{\alpha}\) are minimizers of \(F^{\alpha}_{p,q}\) and \(\tilde{F}^{\alpha}_{p,q}\), the above inequality implies that \[\begin{align} 0&\leq DF^{\alpha}_{p,q}(t_{\alpha})[\tilde{t}_{\alpha}-t_{\alpha}]+D\tilde{F}^{\alpha}_{p,q}(\tilde{t}_{\alpha})[t_{\alpha}-\tilde{t}_{\alpha}] \leq DF_{h,p}(t_{\alpha})[\tilde{t}_{\alpha}-t_{\alpha}] + D\tilde{F}_{h,p}(\tilde{t}_{\alpha})[t_{\alpha}-\tilde{t}_{\alpha}] \\ &=2(\langle \nabla^{g}t_{\alpha}-u,\nabla^{g}(\tilde{t}_{\alpha}-t_{\alpha})\rangle_{H^{p-1}(\Omega)}+\langle \nabla^{g}\tilde{t}_{\alpha}-\tilde{u},\nabla^{g}(t_{\alpha}-\tilde{t}_{\alpha})\rangle_{H^{p-1}(\Omega)})\\ &=2(\langle u-\tilde{u},\nabla^{g}(t_{\alpha}-\tilde{t}_{\alpha})\rangle_{H^{p-1}(\Omega)}-\|\nabla^{g}(t_{\alpha}-\tilde{t}_{\alpha})\|_{H^{p-1}(\Omega)}^{2})\;, \end{align}\] where in third line we used expression 9 expressed in terms of the Sobolev inner product. Combined with the Cauchy-Schwarz inequality results in \[\begin{align} \|\nabla^{g}(t_{\alpha}-\tilde{t}_{\alpha})\|_{H^{p-1}(\Omega)}^{2}&\leq\langle u-\tilde{u},\nabla^{g}(t_{\alpha}-\tilde{t}_{\alpha})\rangle_{H^{p-1}(\Omega)}\\ &\leq \|u-\tilde{u}\|_{H^{p-1}(\Omega)} \|\nabla^{g}(t_{\alpha}-\tilde{t}_{\alpha})\|_{H^{p-1}(\Omega)}\;. \end{align}\] Applying now the Poincaré-Wirtinger inequality (both \(t_{\alpha}\) and \(\tilde{t}_{\alpha}\) satisfy the zero-mean condition) as in the coercivity proof in Lemma 8 gives the desired inequality with a constant which might depend on \(p\), \(\Omega\), \(g\) and \(h\) but not on \(q, \alpha, u\) or \(\tilde{u}\). ◻

Proof. In the first place, the claim holds directly for the \(L^2\) norms by \(C^{0}\) convergence of \((h_{k})_{k\in\mathbb{N}}\): for sufficiently large \(k\in\mathbb{N}\) there exist constants \(a,b>0\) (independent of \(k\)) such that \(a h\leq h_{k}\leq bh\) (analogous bounds hold for the associated volume forms and the tensor norms \(|\cdot|_{h}\) and \(|\cdot|_{h_{k}}\)). Hence, there exist \(k\)-independent constants \(C,C'\) such that for sufficiently large \(k\) and square integrable tensor field \(T\) \[\begin{align} \label{eq:L2comparison} C'\|T\|_{L^{2}(\Omega,h_{k})}\leq\| T\|_{L^{2}(\Omega,h)}\leq C\|T\|_{L^{2}(\Omega,h_{k})} \;. \end{align}\tag{21}\] and, in particular, it holds for \(T=(\nabla^{h_{k}})^{j} f\) with \(j\in\{0,\ldots,p\}\). Moreover, as \(h_{k}\to h\) strongly in \(C^{p}\), the following bounded operator \[\begin{align} \label{eq:difference95derivatives} (\nabla^{h})^{j}-(\nabla^{h_{k}})^{j}: H^{p}(\Omega,h)\to H^{p-j}(\Omega,h)\;, \end{align}\tag{22}\] where \(j\in\{1,\ldots,p\}\), is a differential operator whose coefficients converge to \(0\) in the \(C^{p-j}\) norm and thus in the operator norm. So for any \(f\in H^{p}(\Omega,h)\) and \(j\in\{1,\ldots,p\}\) \[\begin{align} \|(\nabla^{h})^{j}f-(\nabla^{h_{k}})^{j}f\|_{L^{2}(\Omega,h)}\leq\|((\nabla^{h})^{j}-(\nabla^{h_{k}})^{j})f\|_{H^{p-j}(\Omega,h)}\leq \varepsilon_{k}\|f\|_{H^{p}(\Omega,h)}\;, \end{align}\] where \(\varepsilon_{k}\to0\) as \(k\to\infty\). By the triangle inequality and the second inequality in 21 , \[\begin{align} \| (\nabla^{h})^{j}f\|_{L^{2}(\Omega,h)}&\leq \|(\nabla^{h_{k}})^{j}f\|_{L^{2}(\Omega,h)}+\|(\nabla^{h})^{j}f-(\nabla^{h_{k}})^{j}f\|_{L^{2}(\Omega,h)}\\ &\leq C\| (\nabla^{h_{k}})^{j}f\|_{L^{2}(\Omega,h_{k})}+\varepsilon_{k}\|f\|_{H^{p}(\Omega,h)}\;. \end{align}\] Squaring and summing over \(j=1,\ldots, p\) yields: \[\begin{align} \|f\|_{H^{p}(\Omega,h)}\leq C\|f\|_{H^{p}(\Omega,h_{k})}+C'\varepsilon_{k}\|f\|_{H^{p}(\Omega,h)}\;. \end{align}\] Hence, for sufficiently large \(k\) (in particular, when \(\varepsilon_{k}C'<1\)) there exists a \(k\)-independent constant \(C\) such that for any \(f\in H^{p}(\Omega,h)\) \[\begin{align} \|f\|_{H^{p}(\Omega,h)}\leq C\|f\|_{H^{p}(\Omega,h_{k})}\;. \end{align}\] On the other hand, a similar argument exploiting the first inequality in 21 gives \[\begin{align} \| (\nabla^{h_{k}})^{j}f\|_{L^{2}(\Omega,h_{k})}&\leq C\| (\nabla^{h_{k}})^{j}f\|_{L^{2}(\Omega,h)}\leq C\|(\nabla^{h})^{j}f\|_{L^{2}(\Omega,h)}+\varepsilon_{k}\|f\|_{H^{p}(\Omega,h)}\;, \end{align}\] which readily implies the first inequality in ?? . Finally, the same proof holds if one considers a \(H^{p-1}\) regular vector field instead of a function. ◻

Proof. The proof is divided in three parts: first, boundedness of the sequence of minimizers \((t_{k})_{k\in\mathbb{N}}\) is established, then \(C^{r,\gamma}\)-convergence to the minimizer \(t_{\alpha}\) is proven, and, finally, this is upgraded to \(H^{p}\)-convergence.

\(C^{p}\) convergence of \((g_{k})_{k\in \mathbb{N}}\) and \((u_{k})_{k\in \mathbb{N}}\) implies \(C^{p}\) convergence of the sequence of Riemannian metrics \((h_{k})_{k\in\mathbb{N}}\). Let \(\tau\) be a smooth temporal function in \((M,g)\), \(t_{k}\) the unique minimizer of \(F^{\alpha,k}_{p,q}\) on \(\mathcal{T}^{p,k}\) and \(t_{\alpha}\) the minimizer of \(F^{\alpha}_{p,q}\). Up to an additive reparametrization guaranteeing that \(\tau\) and \(t_{\alpha}\) satisfy the zero-mean condition with respect to the measure \(\mu_{h_{k}}\) and with a slight abuse of notation we have that \(\tau,t_{\alpha}\in \mathcal{T}^{p,k}\).

The sequence \((\nabla^{g_{k}}t_{k})_{k\in\mathbb{N}}\) can be bounded by a \(k\)-independent constant \(C\): \[\begin{align} \label{eq:boundtk} \|\nabla^{g_{k}}t_{k}\|_{H^{p-1}(\Omega,h_{k})} \leq C (1+(F^{\alpha,k}_{p,q}(\tau))^{1/2})<C'\;. \end{align}\tag{23}\] The first inequality follows from compactness of \(\Omega\) and \(C^{p}\) convergence of \(u_{k}\) \[\begin{align} \|\nabla^{g_{k}}t_{k}\|_{H^{p-1}(\Omega,h_{k})}&\leq \|u_{k}\|_{H^{p-1}(\Omega,h_{k})}+\|u_{k}-\nabla^{g_{k}}t_{k}\|_{H^{p-1}(\Omega,h_{k})}\leq C(1+ (F^{\alpha,k}_{p,q}(t_{k}))^{1/2}). \end{align}\] together with minimality of \(t_{k}\). It remains to prove the second inequality in 23 . Since \(g_{k}\to g\) uniformly, also \(\nabla^{g_{k}}\tau\to \nabla^{g}\tau\) uniformly and \(|\nabla^{g_{k}}\tau|_{g_{k}}\to |\nabla^{g}\tau|_{g}\) pointwise in \(\Omega\). As \(|\nabla^{g}\tau|_{g}\geq\delta\) for some \(\delta>0\), there exists a \(K\in\mathbb{N}\) with \(|\nabla^{g_{k}}\tau|_{g_{k}}\geq \delta/2\) for all \(k\geq K\). Since for sufficiently large \(k\), \(h_{k}\leq bh\), with \(b>0\), we have that \[\begin{align} \label{eq:bound95tau} &F_{g_{k},q}(\tau)=\int_{\Omega}\frac{1}{|\nabla^{g_{k}}\tau|_{g_{k}}^{2q}}d\mu_{h_{k}}\leq C\int_{\Omega} (\delta/2)^{-2q}d\mu_{h}\leq C'\;. \end{align}\tag{24}\] Finally, by Lemma 23 (applied to \(u_{k}-\nabla^{g_{k}}\tau\)) and \(C^{p}\) convergence of \((g_{k})_{k\in\mathbb{N}}\) and \((u_{k})_{k\in\mathbb{N}}\) also \(F_{h_{k},p}(\tau)\) can be bounded by a \(k\) independent constant, concluding the proof of 23 . Boundedness of \((\nabla^{g_{k}}t_{k})_{k\in\mathbb{N}}\) in \(H^{p-1}(\Omega,h_{k})\) can be upgraded to boundedness in \(H^{p}(\Omega,h)\) of the sequence \((t_{k})_{k\in\mathbb{N}}\): \[\begin{align} \|t_{k}\|_{H^{p}(\Omega,h)}\leq C\|t_{k}\|_{H^{p}(\Omega,h_{k})}\leq C\|\nabla^{h_{k}}t_{k}\|_{H^{p-1}(\Omega,h_{k})}\leq C\|\nabla^{g_{k}}t_{k}\|_{H^{p-1}(\Omega,h_{k})}\leq C'\;, \end{align}\] where the first inequality is due to Lemma 23, the second one to the Poincaré-Wirtinger inequality (applied on \(L^{2}(\Omega,h_{k})\)), the third one to the \(C^{p}\) convergence of the Riemannian metrics and compactness of \(\Omega\), and the final one is 23 . By Banach-Alaoglu there exists a weakly converging subsequence in \(H^{p}(\Omega)\) which, by the Sobolev embedding theorem (after passing to a further subsequence), also converges in the \(C^{r,\gamma}\)-norm to \(t_{\ast}\). Hence, uniform convergence of the gradients implies that \(\nabla^{g}t_{\ast}\) is a.e. causal and satisfies the zero-mean condition (so \(t_{\ast}\in \mathcal{T}^{p}\)). Indeed, as \(d\mu_{h_{k}}=\rho_{k}d\mu_{h}\) with \(\rho_{k}\) positive and continuous and converging uniformly to 1, \(t_{k}\rho_{k}\) converges in \(L^{1}\) to \(t_{\ast}\) and thus \(\int_{\Omega}t_{\ast}d\mu_{h}=\lim_{k\to\infty}\int_{\Omega}t_{k}\rho_{k}d\mu_{h}=0\).

The following inequality plays an important role in order to show that \(t_{\ast}\) minimizes the functional \(F^{\alpha}_{p,q}\): \[\begin{align} \label{eq:liminf95Ftk} F^{\alpha}_{p,q}(t_{\ast})\leq \liminf_{k\to\infty}F_{p,q}^{\alpha,k}(t_{k})\;. \end{align}\tag{25}\] We proceed to prove this expression. In the first place, uniform convergence of \(|\nabla^{g_{k}}t_{k}|_{g_{k}}\) to \(|\nabla^{g}t_{\ast}|_{g}\), implies convergence of the non-negative functions \(f_{k}:=(|\nabla^{g_{k}}t_{k}|_{g_{k}})^{-2q}\rho_{k}\) to the extended real valued function \(f:=(|\nabla^{g}t_{\ast}|_{g})^{-2q}\) (a priori, \(f\) might diverge since the gradient of \(t_{\ast}\) can be null or vanish). By Fatou’s lemma: \[\begin{align} F_{g,q}(t_{\ast})=\int_{\Omega}\frac{1}{|\nabla^{g}t_{\ast}|_{g}^{2q}}d\mu_{h}\leq \liminf_{k\to\infty} \int_{\Omega}f_{k}d\mu_{h}= \liminf_{k\to\infty}F_{g_{k},q}(t_{k})\;. \end{align}\] In particular, the right hand side is bounded by \(F_{p,q}^{\alpha,k}(\tau)\) (since \(t_{k}\) is the minimizer) and thus the gradient of \(t_{\ast}\) is a.e. timelike. On the other hand, since \(X_{k}:=u_{k}-\nabla^{g_{k}}t_{k}\) converges \(H^{p-1}\)-weakly to \(X:=u-\nabla^{g}t_{\ast}\), also \((\nabla^{h_{k}})^{j}X_{k}\) converges weakly in \(L^{2}\) to \((\nabla^{h})^{j}X\) for all \(j\leq p-1\). Weak lower semicontinuity of the \(L^{2}\) norm yields \[\begin{align} F_{h,p}(t_{\ast})&=\sum_{i=0}^{p-1}\int_{\Omega} |(\nabla^{h})^{i}X|_{h}^{2}d\mu_{h}\leq \sum_{i=0}^{p-1}\liminf_{k\to\infty} \int_{\Omega} |(\nabla^{h_{k}})^{i}X_{k}|^{2}_{h}d\mu_{h}\\ &\leq\sum_{i=0}^{p-1}\liminf_{k\to\infty} \int_{\Omega} |(\nabla^{h_{k}})^{i}X_{k}|^{2}_{h_{k}}d\mu_{h_{k}}\leq \liminf_{k\to\infty}F_{h_{k},p}(t_{k})\;, \end{align}\] where the second inequality exploits that for an arbitrary tensor \(T\) and sufficiently large \(k\), \(|T|^{2}_{h}\leq (1+\varepsilon_{k})|T|^{2}_{h_{k}}\rho_{k}\) with \(\varepsilon_{k}\to0\) as \(k\to\infty\). This concludes the proof of expression 25 which, together with the minimality of \(t_{k}\), implies that \[\begin{align} \label{eq:convergence95seq} F^{\alpha}_{p,q}(t_{\ast})\leq \liminf_{k\to\infty}F_{p,q}^{\alpha,k}(t_{k})\leq \limsup_{k\to\infty} F_{p,q}^{\alpha,k}(t_{k})\leq \limsup_{k\to\infty} F_{p,q}^{\alpha,k}(t_{\alpha})= F^{\alpha}_{p,q}(t_{\alpha})\;. \end{align}\tag{26}\] and the last equality follows from \(F^{\alpha,k}_{p,q}(t_{\alpha})\to F^{\alpha}_{p,q}(t_{\alpha})\). Since \(t_{\alpha}\) is the unique minimizer of \(F^{\alpha}_{p,q}\), it follows that \(t_{\ast}=t_{\alpha}\). Hence, any subsequence of the sequence \((t_{k})_{k\in\mathbb{N}}\) admits a further subsequence converging \(H^{p}\)-weakly and \(C^{r,\gamma}\)-strongly to the unique minimizer \(t_{\alpha}\) of \(F^{\alpha}_{p,q}\), which implies convergence of the full sequence.

It remains to show strong \(H^{p}\)-convergence of the sequence \((t_{k})_{k\in\mathbb{N}}\). By 26 , \(F^{\alpha,k}_{p,q}(t_{k})\to F^{\alpha}_{p,q}(t_{\alpha})\) so, in particular, \(F_{h_{k},p}(t_{k})\) converges to \(F_{h,p}(t_{\alpha})\). Together with \(C^{p}\) convergence of \(h_{k}\), it implies that the bounded sequence \((\|X_{k}\|_{H^{p-1}(\Omega,h)})_{k\in\mathbb{N}}\) converges to \(\|X\|_{H^{p-1}(\Omega,h)}\). It follows that \(X_{k}\) converges to \(X\) strongly in \(H^{p-1}\) using the scalar product \((\cdot,\cdot)_{H^{p-1}(\Omega,h)}\): \[\begin{align} \|X_{k}-X\|_{H^{p-1}(\Omega,h)}^{2}=\|X_{k}\|_{H^{p-1}(\Omega,h)}^{2}+ \| X\|^{2}_{H^{p-1}(\Omega,h)}-2(X_{k},X)_{H^{p-1}(\Omega,h)}\;, \end{align}\] where the last expression also converges to \(\|X\|_{H^{p-1}(\Omega,h)}^{2}\) by weak \(H^{p-1}\) convergence of \(X_{k}\) to \(X\). Combined with \(H^{1}\) convergence of \((t_{k})_{k\in\mathbb{N}}\) yields strong \(H^{p}\) convergence. ◻

Proof. The strategy of the proof is the following: under the above assumptions, it will be shown that for all \(t\in \mathcal{T}^{p}\) \[\begin{align} t\circ\Phi\in\mathcal{T}^{p}\quad\textrm{and}\quad F^{\alpha}_{p,q}(t\circ\Phi)=F^{\alpha}_{p,q}(t)\;, \end{align}\] which together with uniqueness of the minimizer implies that \(t_{\alpha}=t_{\alpha}\circ\Phi\).
In the first place, note that \[\begin{align} \label{eq:gradient95change1} &\nabla^{g}(t\circ \Phi)=(d\Phi)^{-1}(\nabla^{g}t\circ \Phi)\;. \end{align}\tag{27}\] Indeed for arbitrary \(x\in \Omega\) and \(Y\in T_xM\) the chain rule yields that \[\begin{align} d(t\circ \Phi)|_{x}(Y)&=dt|_{\Phi(x)}(d\Phi|_{x}(Y))=g_{\Phi(x)}(\nabla^{g}t|_{\Phi(x)},d\Phi|_{x}(Y))\\ &=g_{x}((d\Phi|_{x})^{-1}(\nabla^{g}t|_{\Phi(x)}),Y)\;, \end{align}\] where in the last step we simply used that \(\Phi\) is a \(g\)-isometry. Expression 27 then follows from noting that \(d(t\circ \Phi)|_{x}(Y)=g_{x}(\nabla^{g}(t\circ \Phi)|_{x},Y)\).

Using 27 and that isometries preserve the causal character, for any \(t\in \mathcal{T}^{p}\), \(\nabla^{g}(t\circ \Phi)=d\Phi^{-1}(\nabla^{g}t\circ\Phi)\) is again causal. Since \(d\Phi(u)=u\), the linear isometry \(d\Phi\) preserves also time orientation of vectors with the same orientation as \(u\): for a causal vector \(v\) with the same time orientation as \(u\), the same holds for \(d\Phi(v)\) since \[\begin{align} g(u,d\Phi(v))=g(d\Phi(u),d\Phi(v))=g(u,v)<0\;. \end{align}\] Hence, \(\nabla^{g}(t\circ \Phi)\) is past-directed causal. Furthermore, since \(\Phi^{\ast}g=g\) and \(d\Phi(u)=u\), \(\Phi\) is also an isometry with respect to the Riemannian metric \(h\) and thus preserves the measure \(\mu_{h}\), i.e. \(\Phi_{\ast}\mu_{h}=\mu_{h}\). As \(\Phi\) preserves the subset \(\Omega\) (i.e. \(\Phi(\Omega)=\Omega\)), it follows that \(t\circ \Phi\) also satisfies the zero-mean condition \[\begin{align} \int_{\Omega}(t\circ \Phi )d\mu_{h}=\int_{\Phi(\Omega)}t \;d(\Phi_{\ast}\mu_{h})=\int_{\Omega}t\;d\mu_{h}=0\;. \end{align}\] Hence, \(t\circ\Phi\in \mathcal{T}^{p}\) for any \(t\in \mathcal{T}^{p}\).

On the other hand, expression 27 combined with \((d\Phi)^{-1}(u\circ \Phi)=u\) and that \(\Phi\) is an \(h\)-isometry implies that the Riemannian functional \(F_{h,p}\) remains invariant, \[\begin{align} |u-\nabla^{g}(t\circ\Phi)|_{h}(x)&=|(d\Phi)^{-1}((u-\nabla^{g}t)\circ \Phi)|_{h}(x)=|u-\nabla^{g}t|_{h}(\Phi(x))\\ |(\nabla^{h})^{i}(u-\nabla^{g}(t\circ\Phi))|^{2}_{h}(x)&=|(\nabla^{h})^{i}(u-\nabla^{g}t)|^{2}_{h}(\Phi(x))\\ F_{h,p}(t\circ \Phi)&=\sum_{j=0}^{p-1}\int_{\Omega} |(\nabla^{h})^{j}(u-\nabla^{g}t)|_{h}^{2}\circ \Phi \;d\mu_{h}=F_{h,p}(t)\;, \end{align}\] where \(x\in \Omega,i\in\mathbb{N}\) and we used that the connection \(\nabla^{h}\) behaves naturally under the pullback \(d\Phi^{-1}\)(i.e. \(\nabla^{h}(d\Phi^{-1}(Y\circ \Phi))=d\Phi^{-1}(\nabla^{h}Y\circ \Phi)\) for any vector field \(Y\)). Finally, also the penalty term \(F_{g,q}\) remains invariant: \[\begin{align} F_{g,q}(t\circ\Phi)&=\int_{\Omega}\frac{1}{|\nabla^{g}(t\circ\Phi)|_{g}^{2q}}d\mu_{h}=\int_{\Omega}\Big(\frac{1}{|\nabla^{g}t|_{g}^{2q}}\circ\Phi\Big)\; d\mu_{h}=F_{g,q}(t)\;. \end{align}\] Hence, for any \(t\in\mathcal{T}^{p}\), we have that \[\begin{align} F^{\alpha}_{p,q}(t\circ \Phi)=F^{\alpha}_{p,q}(t) \;. \end{align}\] Uniqueness of the minimizer \(t_{\alpha}\) implies that \(t_{\alpha}\circ\Phi=t_{\alpha}\). ◻

Proof. The first claim simply follows from the fact that for each \(s\in\mathbb{R}\), the isometry \(\Phi_{s}\) preserves the minimizer \(t_{\alpha}\), i.e. \(t_{\alpha}\circ \Phi_{s}=t_{\alpha}\) a.e.. Thus, in the weak sense \[\begin{align} X(t_{\alpha})=\frac{d}{ds}(t_{\alpha}\circ\Phi_{s})\Big|_{s=0}=0\;. \end{align}\] Secondly, assume that \(G\) acts transitively on a non-empty open subset \(U\subset \Omega\) (so \(\mu_{h}(U)>0\)). Let \(p>n/2\), for which there exists a continuous and unique minimizer \(t_{\alpha}\in \mathcal{T}^{p,t}\) of \(F^{\alpha}_{p,q}\) for any \(\alpha>0\). Then, for any \(x,y\in U\) there exists an isometry \(\Phi\in G\) such that \(\Phi(x)=y\) which implies that \(t_{\alpha}\) is constant in \(U\) since \[\begin{align} t_{\alpha}(y)=t_{\alpha}(\Phi(x))=t_{\alpha}(x)\;. \end{align}\] In particular, it follows that \(\nabla^{g}t_{\alpha}=0\) a.e. in \(U\) contradicting that \(\nabla^{g}t_{\alpha}\) is a.e. timelike for any \(\alpha>0\).

Finally, let \(x\in \Omega\) be a fixed point of \(\Phi\). Then, \(u_{x}\) is a fixed timelike vector of \(d\Phi_{x}\). For \(p> n/2+1\), the gradient of the minimizer \(\nabla^{g}t_{\alpha}\) is continuous which together with \(t_{\alpha}\circ \Phi=t_{\alpha}\) in \(\Omega\) and expression 27 proves the claim \[\begin{align} &\nabla^{g}t_{\alpha}(x)=\nabla^{g}(t_{\alpha}\circ\Phi)(x)=(d\Phi|_{x})^{-1}(\nabla^{g}t_{\alpha}(\Phi(x)))=(d\Phi|_{x})^{-1}(\nabla^{g}t_{\alpha}(x))\\ &\implies d\Phi|_{x}(\nabla^{g}t_{\alpha}(x))=\nabla^{g}t_{\alpha}(x)\;, \end{align}\] i.e. \(\nabla^{g}t_{\alpha}(x)\) is a fixed point of \(d\Phi_{x}\). If \(\alpha>0\), \(p>n/2+1+\gamma\), \(q\geq n/\gamma\) and \(u_{x}\neq(\nabla^{g}t_{\alpha})(x)\), then \(d\Phi_{x}\) has even two distinct fixed past-directed timelike vectors. ◻

Proof. As discussed in the proof of the previous corollary, since \(G\) acts transitively on \(\Omega_{+}\) and \(\Omega_{-}\) the minimizer is constant on these subsets. In particular, we denote by \(c_{-},c_{+}\in\mathbb{R}\) its value on \(\Omega_{-}\) and \(\Omega_{+}\) respectively. Let \(\gamma:[0,1]\to\Omega\) be a maximal future-directed causal curve. By assumption, its endpoints are on \(\Omega_{\pm}\) so, without loss of generality, and since \(t_{\alpha}\) is a time function \((t_{\alpha}\circ \gamma)(0)=c_{-}<c_{+}=(t_{\alpha}\circ \gamma)(1)\). As \(t_{\alpha}\circ \gamma\) is strictly increasing and continuous, it takes any value in \((c_{-},c_{+})\) exactly once. So each hypersurface \(\{x\in \Omega:t_{\alpha}(x)=c\}\cap\Omega^{\circ}\) with \(c\in(c_{-},c_{+})\) is intersected exactly once by maximal causal curves in \(\Omega\). Finally, every point in \(x\in \Omega\) is reached by a maximal causal curve in \(\Omega\) (with endpoints on \(\Omega_{\pm}\)), so \(t_{\alpha}(x)\in(c_{-},c_{+})\) and thus the interior level sets \(\{x\in \Omega:t_{\alpha}(x)=c\}\cap \Omega^{\circ}\) with \(c\in(c_{-},c_{+})\) cover \(\Omega^{\circ}\). ◻