Isometries preserve maximality and transported boundary

Write ${\varphi }_{\ast }U=d\varphi \circ U\circ {\varphi }^{-1}$ and define the pulled-back connection by $D_{U}V={\varphi }_{\ast }^{-1}({\mathrm{\nabla }}_{{\varphi }_{\ast }U}{\varphi }_{\ast }V)$. The connection rules and metric compatibility pull back through the isometry, so $D$ is a metric-compatible connection for $g$. By bracket preservation its torsion is $D_{U}V-D_{V}U-[U,V]={\varphi }_{\ast }^{-1}({\mathrm{\nabla }}_{{\varphi }_{\ast }U}{\varphi }_{\ast }V-{\mathrm{\nabla }}_{{\varphi }_{\ast }V}{\varphi }_{\ast }U-[{\varphi }_{\ast }U,{\varphi }_{\ast }V])=0$. Uniqueness therefore gives $D=\mathrm{\nabla }$.

Give $\varphi (\widehat{\mathrm{\Sigma }})$ the transported smooth structure: if $x$ is a boundary or interior chart on $\widehat{\mathrm{\Sigma }}$, its image chart is $x\circ {\varphi }^{-1}$. Its inclusion into $M$ is the original embedding followed by the ambient diffeomorphism $\varphi$, so it is a smooth embedding, and the restriction $\varphi :\widehat{\mathrm{\Sigma }}\to \varphi (\widehat{\mathrm{\Sigma }})$ is a diffeomorphism of manifolds with boundary. Intrinsic boundary preservation now gives $\mathrm{\partial }(\varphi (\widehat{\mathrm{\Sigma }}))=\varphi (\mathrm{\partial }\widehat{\mathrm{\Sigma }})=\varphi (S)$.

For the specialization let ${\mathrm{\Phi }}_t$ be the flow of the complete stationary field $T$. It is global and smooth, and ${\mathrm{\Phi }}_{t+h}={\mathrm{\Phi }}_h\circ {\mathrm{\Phi }}_t$ with ${\mathrm{\Phi }}_t^{-1}={\mathrm{\Phi }}_{-t}$, so each ${\mathrm{\Phi }}_t$ is a diffeomorphism. Differentiating ${\mathrm{\Phi }}_{t+h}^{\ast }g={\mathrm{\Phi }}_t^{\ast }({\mathrm{\Phi }}_h^{\ast }g)$ at $h=0$ gives $\frac{d}{dt}{\mathrm{\Phi }}_t^{\ast }g={\mathrm{\Phi }}_t^{\ast }({\mathcal{L}}_{T}g)=0$. For fixed $p,U,V$ each evaluation $({\mathrm{\Phi }}_t^{\ast }g{)}_p(U,V)$ is a smooth real function of $t$. The mean value theorem on the interval joining $0$ and $t$ makes it constant, hence ${\mathrm{\Phi }}_t^{\ast }g=g$ for every $t$. This calculation does not require $T$ to be nonzero.

Put ${\mathrm{\Sigma }}^{\mathrm{\prime }}=\varphi (\widehat{\mathrm{\Sigma }})$. The differential takes each tangent space onto that of the image, and metric invariance makes the image spacelike. Since $\varphi$ preserves time orientation, $d\varphi (n)$ is its future unit normal $n^{\mathrm{\prime }}$. Consequently $h^{\mathrm{\prime }}(d\varphi U,d\varphi V)=h(U,V)$ and $k^{\mathrm{\prime }}(d\varphi U,d\varphi V)=-g({\mathrm{\nabla }}_{d\varphi U}{n}^{\mathrm{\prime }},d\varphi V)=-g({\mathrm{\nabla }}_{U}n,V)=k(U,V)$. Tracing in an $h$-orthonormal basis and its $d\varphi$ image gives equal metric traces, so maximality is preserved. These identities hold up to the boundary with the supplied smooth structure.

Fix $p$ and define $w_s=(d{\mathrm{\Phi }}_s{\mathrm{\mid }}_p{)}^{-1}{N}_{{\mathrm{\Phi }}_s(p)}\in T_{p}M$. Smoothness and the isometry identity make $s\mapsto w_s$ a continuous timelike path, with $w_0=N_p$. It stays in the future component of the timelike set of $T_{p}M$. The linear isometry $d{\mathrm{\Phi }}_s{\mathrm{\mid }}_p$ permutes the two timelike components and maps $w_s$ to the future vector $N_{{\mathrm{\Phi }}_s(p)}$; it therefore maps the future component onto the future component. Thus every ${\mathrm{\Phi }}_s$ preserves time orientation. Applying steps 2.1 and 1.2 to $\varphi ={\mathrm{\Phi }}_s$, and using the hypothesized setwise invariance ${\mathrm{\Phi }}_s(S)=S$, proves the claimed boundary and maximality assertions for every translate.