Mean-curvature variation and the stationary lapse equation

First suppose $V=an$. In local coordinates $x^i$ on $\mathrm{\Sigma }$, put $e_i={\mathrm{\partial }}_i{F}_s$ and $K_{ij}=g({\mathrm{\nabla }}_{e_i}n,e_j)=-k_{ij}$ at $s=0$. Differentiating $g(n,n)=-1$ and $g(n,e_i)=0$, using torsion-freeness and $[{\mathrm{\partial }}_s,e_i]=0$, gives ${\mathrm{\nabla }}_{{\mathrm{\partial }}_s}n={\mathrm{grad}}_{h}a$ at $s=0$. Metric compatibility also gives ${\dot{h}}_{ij}=2a{K}_{ij}$ and ${\dot{h}}^{ij}=-2a{K}^{ij}$.

Commuting the covariant derivatives by the curvature convention and differentiating $K_{ij}$ gives ${\dot{K}}_{ij}={\mathrm{\nabla }}_i^h{\mathrm{\nabla }}_j^{h}a+ag(R(n,e_i)n,e_j)+a{K}_i{}^p{K}_{pj}$, where ${\mathrm{\nabla }}^h$ is the Levi–Civita connection of $h$. Using ${\mathrm{tr}}_h{\mathrm{Hess}}_{h}a={\mathrm{\Delta }}_{h}a$ from F5, tracing this identity and including ${\dot{h}}^{ij}{K}_{ij}$ yields $\dot{H}={\mathrm{\Delta }}_{h}a-a\mathrm{\mid }k{\mathrm{\mid }}_h^2+a{h}^{ij}g(R(n,e_i)n,e_j)$. The displayed Ricci convention and the curvature symmetries of the metric-compatible torsion-free connection give $h^{ij}g(R(n,e_i)n,e_j)=-\mathrm{Ric}(n,n)$, proving the formula for a normal variation.

For a general variation, locally reparametrize the domain by a time-dependent diffeomorphism whose initial velocity cancels $X$. The resulting variation has initial velocity $an$; differentiating the scalar pullback under the inverse reparametrization adds $X(H)$. This proves the stated formula for arbitrary $V$; the graph mean-curvature map uses the same pullback convention.

By F3 the local flow of a Killing field consists of local isometries. Transporting $\mathrm{\nabla }$ by a local isometry gives a torsion-free metric-compatible connection, which equals $\mathrm{\nabla }$ by uniqueness in F4. The flow starts at the identity and hence preserves future direction on its local flow domain. It therefore carries the future normal and mean curvature of $\mathrm{\Sigma }$ to those of the image hypersurface. Its pulled-back mean curvature is therefore constant in the flow parameter. Its variational field is $T=Nn+X$; if $H=0$, then $X(H)=0$, and the variation formula gives the stated lapse equation. The calculation is local and imposes no boundary or asymptotic condition.