Anchored lapse propagation and quantitative stationary coordinate transfer

The operator and curvature conventions are [F4]. By [F1], $\mathrm{\mid }{k}_j\mathrm{\mid }$ and the coordinate normal components are bounded. The Ricci components in [F4] depend smoothly on the ambient metric, its inverse and its coordinate derivatives through order two; joint smoothness and compactness of the background parameter-position set bound these components uniformly. Thus the contraction with the bounded normal components is bounded, and $V_j$ is uniformly bounded. The local induced metric coefficients and their required derivatives have uniform bounds and common ellipticity in the buffered charts of [F1]. Consequently $m_j=\sqrt{\det ((h_j{)}_{ab})}$ has common positive lower and upper bounds, $A_j=(m_j{h}_j^{ab})$ has common positive ellipticity and upper bounds, and $q_j=m_j{V}_j$ has a common nonnegative upper bound. The common radii, metric bounds and buffers in [F1] allow its bounded connected cover of $L_j$ to be covered by coordinate half-balls with uniformly bounded cardinality, each inside a larger controlled coordinate ball; their overlap graph along $L_j$ is connected. Indeed one can take the small intrinsic covering balls in [F1] inside these coordinate half-balls by the lower metric bound, retaining their centers and witnessed overlaps. The larger coordinate balls have a common radius chosen below the chart buffer. On each such ball the smooth positive lapse is a $C^2$ pointwise solution with the nonnegative bounded potential specified above. The inherited choice hypothesis in [F2] holds, so [F2] gives a common constant $C\ge 1$ comparing any two lapse values in each half-ball.

Let $J$ be a common upper bound for the number of these covering half-balls and choose the anchor $x_{\ast }\in L_j$ with $N_j(x_{\ast })\ge a$. For $x\in L_j$, take covering half-balls containing $x_{\ast }$ and $x$. A path in the connected overlap graph can have repeated vertices deleted, leaving at most $J$ vertices. At each successive overlap choose a point in $L_j$ lying in the two half-balls. Comparing consecutive values, from $x_{\ast }$ through those points to $x$, uses at most $J$ of the local inequalities of step 1.1. Multiplication yields $N_j(x_{\ast })\le C^J{N}_j(x)$, hence $N_j(x)\ge a\mathrm{/}{C}^J$. Put $H=C^J$. This uses an actual anchor, not an individual asymptotic limit alone; the finite choices and multiplication introduce no new analytic premise.

Let $\pi (t,x)=x$ be stationary projection and parametrize a local temporal graph near a point $p\in L_j$ by coordinates $y$. The determinant of $D(\pi {\mathrm{\mid }}_{{\mathrm{\Sigma }}_j})$ is, up to smooth bounded nonzero background and graph-volume factors, $-g_j(T,n_j)=N_j$: compare the ambient volume of $(T,{\mathrm{\partial }}_{y^1},{\mathrm{\partial }}_{y^2},{\mathrm{\partial }}_{y^3})$ with the product volume. The local tangent vectors and background coordinate changes have common upper bounds by [F1]. Step 2.1 therefore gives a common lower absolute determinant bound at $p$; the cofactor formula and the upper matrix-entry bounds give a common inverse-matrix bound there. In a fixed base chart write $F_j(y)=\pi ({\iota }_j(y))$, where ${\iota }_j$ parametrizes the temporal graph and ${\iota }_j(0)=p$. The map is smooth on an open chart domain and its derivative at zero is invertible, so [F5] supplies a smooth local inverse $G_j$ with $D{G}_j=(D{F}_j\circ G_j{)}^{-1}$. For each $r\ge 2$, differentiating $F_j\circ G_j=\mathrm{id}$ $r$ times gives $D{F}_j(G_j)D^r{G}_j+R_{j,r}=0$, where $R_{j,r}$ is a finite sum of products of derivatives of $F_j$ through order $r$ and of $G_j$ through order $r-1$. Thus uniform bounds for these derivatives follow recursively wherever the inverse-matrix bound holds.

Choose a fixed finite base atlas near $\pi (Q)$, with relatively compact smaller chart regions covering $\pi (Q)$ inside larger regions, including the fixed buffered chart pieces used for $K$. Carry out the following construction in each such chart for every center whose projection is in its closed smaller region, and take common constants over this finite collection. By [F1] the centered temporal domains contain a common closed coordinate ball ${\bar{B}}_{\rho }(0)$ inside their buffers, after shrinking $\rho >0$; their projections lie in the larger base charts. Let $B_0\ge 1$ bound $\mathrm{\parallel }D{F}_j(0{)}^{-1}\mathrm{\parallel }$ from step 3.1 and let $M_2$ bound $\mathrm{\parallel }{D}^2{F}_j\mathrm{\parallel }$ on the buffered domains. Shrink $\rho$ so that $B_0{M}_2\rho \le 1\mathrm{/}2$. The mean-value inequality in [F6], applied first to $D{F}_j$ on a containing convex buffered ball and then to $y\mapsto F_j(y)-D{F}_j(0)y$ on the open radius-$\rho$ ball, gives $\mathrm{\mid }{F}_j(y)-F_j(z)-D{F}_j(0)(y-z)\mathrm{\mid }\le M_2\rho \mathrm{\mid }y-z\mathrm{\mid }$, hence $\mathrm{\mid }{F}_j(y)-F_j(z)\mathrm{\mid }\ge (2B_0{)}^{-1}\mathrm{\mid }y-z\mathrm{\mid }$. Continuity extends this inequality to the closed ball. The same derivative estimate gives invertibility of $D{F}_j(y)$ and $\mathrm{\parallel }D{F}_j(y{)}^{-1}\mathrm{\parallel }\le 2B_0$ throughout the ball. Set $\epsilon =\rho \mathrm{/}(4B_0)$. For any base point $b$ with $\mathrm{\mid }b-F_j(0)\mathrm{\mid }<\epsilon$, the function $\mathrm{\mid }{F}_j(y)-b{\mathrm{\mid }}^2$ attains a minimum on the nonempty closed bounded ball by [F6]. On its boundary, $\mathrm{\mid }{F}_j(y)-b\mathrm{\mid }\ge \rho \mathrm{/}(2B_0)-\mathrm{\mid }b-F_j(0)\mathrm{\mid }>\mathrm{\mid }b-F_j(0)\mathrm{\mid }$, so a minimizing point is interior. At an interior minimum, each coordinate difference quotient has nonnegative right and nonpositive left limits, so all first derivatives vanish, giving $D{F}_j(y{)}^T(F_j(y)-b)=0$; invertibility forces $F_j(y)=b$. The segment inequality gives uniqueness, and [F5] makes this inverse smooth. Thus the inverse exists on the whole base ball $B_{\epsilon }(F_j(0))$, with inverse-matrix bound $2B_0$. The recursion in step 3.1 gives every fixed finite derivative bound on these common-radius balls. All constants are independent of j and the center $p\in L_j$.

Fix a compact base set $K$ satisfying the lifted-containment condition in the Statement. Every point of its graph lies in $L_j$, so step 4.1 applies at every such point. The stationary time coordinate composed with ${\iota }_j\circ G_j$ is the existing stationary height $u_j$ on each inverse ball; global graph uniqueness makes these expressions agree. The time coordinate and every fixed derivative of the buffered temporal parametrizations have common bounds by [F1]. In particular $\mathrm{\mid }{u}_j\mathrm{\mid }\le {\sup }_Q\mathrm{\mid }t\mathrm{\mid }<\mathrm{\infty }$ on $K$, and the graph lifts of the inverse balls remain in $Q_{+}$, where $\mathrm{\mid }t\mathrm{\mid }\le {\sup }_{Q_{+}}\mathrm{\mid }t\mathrm{\mid }<\mathrm{\infty }$; these bounds follow by a finite cover of the compact tubes by bounded product-coordinate boxes. Composition and the inverse derivative bounds therefore give uniform derivatives of every fixed finite order on the common-radius base balls. In each fixed base chart ${\varphi }_{\mathrm{\ell }}$, pairs $z,w\in K_{\mathrm{\ell }}$ with $\mathrm{\mid }z-w\mathrm{\mid }<\epsilon \mathrm{/}2$ are joined by a straight segment in the inverse ball centered at the first point; the whole ball lies in the larger chart, even when the segment leaves $K_{\mathrm{\ell }}$. The mean-value bound in [F6], applied to the order-$m$ coordinate derivative fields on that convex ball, gives $\mathrm{\mid }{D}^m{v}_{j,\mathrm{\ell }}(z)-D^m{v}_{j,\mathrm{\ell }}(w)\mathrm{\mid }\le C_{m+1}\mathrm{\mid }z-w\mathrm{\mid }$ there. For pairs at distance at least $\epsilon \mathrm{/}2$, the order-$m$ supremum bound gives the difference quotient bound $2C_m(2\mathrm{/}\epsilon {)}^{\alpha }$. These two estimates bound the order-$m$ Holder seminorm for every $0<\alpha <1$, including $m=0$. For every multi-index $\beta$ of order $m$, these estimates apply to $D^{\beta }{v}_{j,\mathrm{\ell }}$, where $v_{j,\mathrm{\ell }}=u_j\circ {\varphi }_{\mathrm{\ell }}^{-1}$; all lower-order suprema are already bounded. Taking the finite sums and maximum in the compact-set formula of the Statement gives the asserted full $C^{m,\alpha }(K)$ bounds and in particular the full $C^{4,\alpha }(K)$ bound. This covers exactly the target graph points in the controlled anchored portions.