Uniform scalar angle inequalities from local background tensors

In [F2], set $Q_a=-g(n,D_{e_a}U)$. By [F1], $\mathrm{\mid }n{\mathrm{\mid }}_e\le \sqrt{2}v$ and $\mathrm{\mid }{e}_a{\mathrm{\mid }}_e\le \sqrt{2}v$, while the metric pairing has operator norm one in an $e$-orthonormal observer frame. Thus $\mathrm{\mid }Q\mathrm{\mid }\le K_1{v}^2$. The other first-derivative term is $kZ$, and [F1] gives $\mathrm{\mid }Z{\mathrm{\mid }}^2=v^2-1$.

The contracted Codazzi term satisfies $\mathrm{\mid }\mathrm{Ric}(n,e_b)\mathrm{\mid }\le CK{v}^2$; multiplying by the three components $Z_b$, whose total norm is at most $v$, bounds its contribution by $CK{v}^3$. The $DU$ term in the Laplacian formula is bounded by $CK\mathrm{\mid }k\mathrm{\mid }{v}^2$. The $D^{2}U$ term has one $n$ and two tangent vector slots, so is bounded by $CK{v}^3$. Consequently $\mathrm{\Delta }v\ge \mathrm{\mid }k{\mathrm{\mid }}^{2}v-CK\mathrm{\mid }k\mathrm{\mid }{v}^2-CK{v}^3$.

Apply the elementary inequality $ab\le \epsilon a^2+b^2\mathrm{/}(4\epsilon )$ to $a=\mathrm{\mid }k\mathrm{\mid }\sqrt{v}$ and $b=CK{v}^{3\mathrm{/}2}$, absorbing constants. This yields the displayed $(1-\epsilon )\mathrm{\mid }k{\mathrm{\mid }}^{2}v-C_{\epsilon }{v}^3$ bound for the given background bound. No bound on the candidate $k$ was assumed or used.

For the family assertion, work in any one of the fixed charts on its compact coordinate set $L_j$. The $C^2$ compact-open continuity gives jointly continuous coordinate components of $g_p,U_p$ and their spatial derivatives through order two on $P\times L_j$: the change in $p$ is controlled uniformly on $L_j$, and for fixed $p$ these components are continuous in the spatial point. Matrix inversion is continuous wherever the determinant is nonzero, by the adjugate formula. Use the Christoffel formula in [F4], with parameter subscripts suppressed for the coordinate calculation. Differentiating $g^{-1}g=I$ gives ${\mathrm{\partial }}_i{g}^{-1}=-g^{-1}({\mathrm{\partial }}_{i}g)g^{-1}$, so $\mathrm{\Gamma }$ and $\mathrm{\partial }\mathrm{\Gamma }$ are finite sums of products of entries of $g^{-1}$ and spatial derivatives of $g$ through order two. Put $A^a{}_j=(D_{j}U{)}^a={\mathrm{\partial }}_j{U}^a+{\mathrm{\Gamma }}^a{}_{jb}{U}^b$. The definition of $D^{2}U$ in [F2] gives $(D^{2}U{)}^a{}_{ij}={\mathrm{\partial }}_i{A}^a{}_j+{\mathrm{\Gamma }}^a{}_{ib}{A}^b{}_j-{\mathrm{\Gamma }}^b{}_{ij}{A}^a{}_b$. Expanding the curvature operator of [F4] on commuting coordinate fields gives $R^a{}_{bij}={\mathrm{\partial }}_i{\mathrm{\Gamma }}^a{}_{jb}-{\mathrm{\partial }}_j{\mathrm{\Gamma }}^a{}_{ib}+{\mathrm{\Gamma }}^a{}_{ic}{\mathrm{\Gamma }}^c{}_{jb}-{\mathrm{\Gamma }}^a{}_{jc}{\mathrm{\Gamma }}^c{}_{ib}$, and ${\mathrm{Ric}}_{bj}={\sum }_a{R}^a{}_{baj}$ is its finite contraction. Thus the components of $D^{(p)}{U}_p$, $(D^{(p)}{)}^2{U}_p$ and $\mathrm{Ric}(g_p)$ use only finite sums and products of those jointly continuous metric entries and spatial derivatives of $U_p$ through order two, and are jointly continuous. Likewise $e_p=g_p+2U_p^{\mathrm{♭}}\otimes U_p^{\mathrm{♭}}$ and its inverse vary continuously; each $e_p$ is positive definite by [F1]. Their finite-dimensional tensor operator norms consequently vary continuously: near any fixed positive metric, its quadratic form is uniformly comparable to that fixed metric, with comparison factors tending to one as its entries converge, and the same holds for the inverse and tensor products. Thus $B_j(p,x)=\mathrm{\mid }{D}^{(p)}{U}_p{\mathrm{\mid }}_{e_p}+\mathrm{\mid }(D^{(p)}{)}^2{U}_p{\mathrm{\mid }}_{e_p}+\mathrm{\mid }\mathrm{Ric}(g_p){\mathrm{\mid }}_{e_p}$ is continuous on $P\times L_j$. By [F3] this product is compact; the metric $\max \{d_P(p,p^{\mathrm{\prime }}),\mathrm{\mid }x-x^{\mathrm{\prime }}\mathrm{\mid }\}$ induces its product topology, since its balls are products of balls. The extreme value theorem in [F3] gives a finite maximum of $B_j$. Take $K_L$ to be the largest of the finitely many such maxima. Steps 1.1–3.1 now apply on $L$ to every family member with this same $K_L$, proving the stated uniformity independently of the candidate.