Stationary maximal graphs and normalized graph heights

PDF p. 12: §3, graph definitions before Theorem 3.1

Let ${\mathrm{\Sigma }}_1$ be the supplied original smooth spacelike maximal section, with interior ${\mathrm{\Sigma }}_1^{\circ }$ and compact boundary $S$, and suppose the exterior is identified with the product $\mathcal{E}={\mathbb{R}}_t\times {\mathrm{\Sigma }}_1^{\circ }$ by the flow of the stationary field $T={\mathrm{\partial }}_t$. The coordinate $t$ is the flow parameter and ${\mathrm{\Sigma }}_1^{\circ }$ is identified with $t=0$. Suppose the Harrison normalization data are supplied on an ambient open set $U$ containing $\mathcal{E}$ and a chosen point $p\in S$, with $T_p=0$ and $d_q=D_q(p)$. For the parameter in question, suppose a Harrison one-form is supplied on an open subset $V\subseteq U$ with $D_q>0$ and $\mathcal{E}\subseteq V$, and let $g_q$ be its Lorentzian metric restricted to $\mathcal{E}$. Thus $d_q$ is evaluated at the ambient boundary point $p$, outside the open product exterior, and is used as a constant on that exterior.

Let $N$ denote the original time-orientation field supplied with the stationary Lorentzian data. Write $A_{q}v=v+D_q^{-1}{W}_q(v)T$ for the smooth invertible bundle map in the Harrison construction, so $g_q=(D_q\mathrm{/}4)A_q^{\ast }g$. We time-orient $g_q$ by declaring a $g_q$-timelike vector $v$ future-directed exactly when $g(A_{q}v,N)<0$.

For $q\in (0,1]$ and a smooth real function $u_q$ on ${\mathrm{\Sigma }}_1^{\circ }$, its stationary graph is ${\mathrm{\Sigma }}_q^{\circ }=\{(u_q(x),x):x\in {\mathrm{\Sigma }}_1^{\circ }\}$. It is a stationary maximal graph when its image is spacelike for $g_q$ and its mean curvature in $g_q$ vanishes. Here spacelike means that the induced metric $h_q$ is positive definite, and mean curvature is $H_q=-{\mathrm{tr}}_{h_q}{k}_q$, where $k_q(A,B)=-g_q({\mathrm{\nabla }}_A^q{n}_q,B)$ for tangent vectors $A,B$, the future unit normal $n_q$ and the Levi–Civita connection ${\mathrm{\nabla }}^q$ of $g_q$. When a smooth hypersurface with boundary ${\mathrm{\Sigma }}_q$ is separately supplied with interior ${\mathrm{\Sigma }}_q^{\circ }$ and boundary $S$, we say that the graph has boundary $S$; this condition does not require $u_q$ to extend continuously to $S$.

On the specified Cartesian end, write $r=\mathrm{\mid }x\mathrm{\mid }$ and introduce $t_q=d_{q}t\mathrm{/}(4q)$, $x_q=qx$, and $r_q=\mathrm{\mid }{x}_q\mathrm{\mid }=qr$. The normalized graph height ${\tilde{u}}_q$ is defined on the rescaled end by ${\tilde{u}}_q(qx)=d_q{u}_q(x)\mathrm{/}(4q)$. The normalization condition is ${\tilde{u}}_q(x_q)\to 0$ as $r_q\to \mathrm{\infty }$, uniformly in angular direction.

For a fixed $\delta \in (0,1)$, an estimate for a family indexed by $q\in [\delta ,1]$ is uniform when its constants and its threshold radius in the stated coordinates can be chosen independently of $q$; they may depend on $\delta$. This convention alone asserts no estimate and gives no uniformity as $\delta \downarrow 0$.

Mass-one convention and Harrison denominator

Stationary Harrison one-form and metric

Stationary Lorentzian data and tensor conventions

Spacelike sections, maximality and lapse–shift data

Differentiated asymptotic bounds and parameter uniformity

Smooth embeddings

Given: The supplied product, section, Harrison data, positive parameter and graph function in the Definition.

Goal: Justify the graph and normalized-height conventions without asserting existence of a maximal section.

1.1 The map $j_q:x\mapsto (u_q(x),x)$ is smooth and injective. Its differential sends a tangent vector $v$ to $(d{u}_q(v),v)$ and is injective because its spatial component is the identity. The product projection is continuous, and its restriction to the image of $j_q$, with the subspace topology, is an inverse of $j_q$. Hence $j_q$ is a homeomorphism onto its image and a smooth immersion, so it is a smooth embedding by [F5]. By [F6], its image, with the smooth structure transported by $j_q$, is a smooth embedded three-dimensional submanifold of the four-dimensional product, hence a hypersurface. Spacelikeness and vanishing mean curvature are explicit conditions on that image; the supplied smooth extension with boundary $S$ is an additional condition. [given, F2, F3, F5, F6]

2.1 Set $Y_q=A_q^{-1}N$. By [F2] and [F7], this is a smooth field with $g_q(Y_q,Y_q)=(D_q\mathrm{/}4)g(N,N)<0$. For every $g_q$-timelike vector $v$, the pullback identity gives $g_q(v,Y_q)=(D_q\mathrm{/}4)g(A_{q}v,N)$, so the future convention in the Definition is precisely the time orientation defined by $Y_q$. The spacelike-hypersurface interface therefore supplies a unique smooth future unit normal on every spacelike graph and makes its stated mean curvature well defined. [F2, F3, F7, step 1.1, algebra]

3.1 At the supplied zero of $T$, $f(p)=0$, so $d_q=(1+q{)}^2+(1-q{)}^2\psi (p{)}^2>0$. Since $q>0$, both coordinate scalings are invertible on the end, and substitution of $t=u_q(x)$ gives exactly $t_q={\tilde{u}}_q(x_q)$. The limit and parameter-uniformity clauses are therefore definite conditions in these coordinates, with the stated dependence of constants. No existence, boundary regularity, or estimate follows merely from the definition. [given, F1, F4, step 1.1, algebra] ∎

The cited article attributes the graph and normalization conventions. The justification uses the declared embedding theorem, the Harrison pullback identity, the original time orientation and local product and coordinate calculations; it does not use the article as a proof of the existence of the graph.