Normal graphs and the nonlinear mean-curvature map

Assume the Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ for the inherited tangent-bundle smooth structures used below.

Let $(M,g,N)$ be a time-oriented smooth Lorentzian four-manifold of signature $(-,+,+,+)$, with metric-selected connection $\mathrm{\nabla }$, and let $\mathrm{\Sigma }\subset M$ be a smooth embedded spacelike hypersurface. Thus $g$ restricts to a positive definite metric on each tangent space of $\mathrm{\Sigma }$. Write $n$ for its future-directed unit normal: $g(n,n)=-1$, $g(n,N)<0$, and $g(n,V)=0$ for every $V\in T\mathrm{\Sigma }$.

Let $Y$ be a smooth vector field on an open neighborhood $U$ of $\mathrm{\Sigma }$, transverse to $\mathrm{\Sigma }$, meaning $Y_x\mathrm{\notin }{T}_x\mathrm{\Sigma }$ at each $x\in \mathrm{\Sigma }$. Let ${\varphi }^Y$ be a local flow of $Y$, with open domain $\mathcal{D}\subseteq \mathbb{R}\times U$. For a smooth real function $w$ on $\mathrm{\Sigma }$ with $(w(x),x)\in \mathcal{D}$ for all $x$, put $F_w(x)={\varphi }^Y(w(x),x)$. When $F_w$ is an embedding and its image ${\mathrm{\Sigma }}_w$ is spacelike, $F_w$ and ${\mathrm{\Sigma }}_w$ are called the flow graph map and flow graph of $w$.

For the normal exponential construction, take a smooth real function $w$ such that $w(x)n_x$ belongs to the exponential map domain at every $x$. Put $E_w(x)={\exp }_x(w(x)n_x)$. When $E_w$ is an embedding with spacelike image, it and its image are the normal exponential graph map and normal exponential graph. In a specified normal tubular neighborhood this construction may be restricted to those $w$ for which $(x,w(x))$ lies in the chosen domain of a diffeomorphism $(x,s)\mapsto {\exp }_x(s{n}_x)$ onto that neighborhood; the existence of such a neighborhood is not asserted here.

For either admissible graph map $G_w$ above, let $n_w$ be the future unit normal to its image, $h_w$ the metric induced there by $g$, and $K_w(V,W)=g({\mathrm{\nabla }}_V{n}_w,W)$ for tangent vectors. At each image point, let $S_w$ be the shape operator determined by $h_w(S_{w}V,W)=K_w(V,W)$; write ${\mathrm{tr}}_{h_w}{K}_w:=\mathrm{tr}(S_w)$, the basis-independent trace of this tangent-space endomorphism. The mean-curvature map assigns to $w$ the function $\mathcal{H}(w)=G_w^{\ast }({\mathrm{tr}}_{h_w}{K}_w)$ on $\mathrm{\Sigma }$. Equivalently, with any local unit timelike extension of $n_w$, this is the pullback of ${\mathrm{div}}_g{n}_w$. A graph is maximal when $\mathcal{H}(w)=0$.

If $S\subseteq \mathrm{\Sigma }$ is a prescribed boundary set, the homogeneous Dirichlet condition is $w{\mathrm{\mid }}_S=0$. It fixes each point of $S$ under either graph map. Boundary regularity, spacelikeness and embedding of a candidate graph, differentiability of $\mathcal{H}$ between Banach spaces, and representation of other hypersurfaces as such graphs require separate hypotheses.

Spacelike hypersurfaces, acausal sets, Cauchy sets, domains of dependence and Cauchy horizons

The future unit normal, induced metric and second fundamental form of a spacelike hypersurface

The geodesic initial-value problem and the exponential map of the metric-selected connection

Local and global flows generated by a vector field

Metric compatibility, torsion freedom and the metric-selected connection class

Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model

The derivative of the metric density and the chart formula for the divergence of a vector field

The basis-independent trace of an endomorphism of a finite-dimensional vector space

The Axiom of Countable Choice (AC_ω)

Smooth embeddings

The image of a smooth embedding is an embedded submanifold

Diffeomorphisms and local diffeomorphisms of manifolds

The local flow and exponential map give smooth graph maps on their stated domains. The inherited ${\mathrm{A}\mathrm{C}}_{\omega }$ convention supplies the tangent-bundle smooth structures, including the one used for smooth dependence on the initial vector in the exponential construction. Embedding and spacelikeness are explicit admissibility conditions, so each image is a spacelike embedded hypersurface. The image of a smooth embedding is an embedded submanifold Its unique smooth future normal and extension-independent tangential derivatives give the smooth tensor $K_w$ and its trace; pulling this scalar back by $G_w$ defines $\mathcal{H}(w)$. In an $h_w$-orthonormal tangent basis, the defining relation for $S_w$ gives $\mathrm{tr}(S_w)={\sum }_{i=1}^3{h}_w(S_w{e}_i,e_i)={\sum }_{i=1}^3{K}_w(e_i,e_i)$; basis independence follows from the endomorphism trace. For a local unit timelike extension of $n_w$, metric compatibility gives $g({\mathrm{\nabla }}_{n_w}{n}_w,n_w)=\frac{1}{2}{n}_w(g(n_w,n_w))=0$. The divergence contraction formula in a local orthonormal frame consisting of $n_w$ and three tangent unit vectors $e_1,e_2,e_3$ gives ${\mathrm{div}}_g{n}_w=-g({\mathrm{\nabla }}_{n_w}{n}_w,n_w)+{\sum }_{i=1}^{3}g({\mathrm{\nabla }}_{e_i}{n}_w,e_i)={\mathrm{tr}}_{h_w}{K}_w$, independent of that unit extension. Finally, the flow and exponential identities at zero imply that $w{\mathrm{\mid }}_S=0$ fixes $S$ pointwise.

The declared library normal lemma supplies the unique smooth future unit normal and extension-independent tangential covariant derivative for a spacelike embedded hypersurface. The future unit normal, induced metric and second fundamental form of a spacelike hypersurface

Gourgoulhon's cited equations record the normal and curvature conventions. His extrinsic curvature has the opposite sign from $K_w$ here; his trace-divergence identity agrees after that sign conversion. The local justification above uses the declared mathematical suppliers.