Conformal construction of constrained initial data — Examples

Given: The flat torus, $L_{\ast }>0$, $a_{\ast }>0$, and a constant $\tau \mathrm{\ne }0$ with the units and coordinate convention in the Example. Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ for the torus supplier and the inherited embedded Lorentzian constraint interpretation.

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), Vacuum conformal constraint data: ${\mathrm{A}\mathrm{C}}_{\omega }$ is the axiom that every countable family of nonempty sets has a choice function; the torus supplier assumes it for its quotient convention, and the conformal-data definition requires it for an embedded Lorentzian interpretation.

Vacuum conformal constraint data: $\tilde{D}$ is the Levi-Civita connection of $\tilde{h}$; TT means ${\tilde{h}}^{ij}{\tilde{\sigma }}_{ij}=0$ and ${\tilde{D}}^i{\tilde{\sigma }}_{ij}=0$; and the ansatz is $h={\varphi }^4\tilde{h}$, $K={\varphi }^{-2}\tilde{\sigma }+(\tau \mathrm{/}3)h$.

Conformal reduction of the vacuum constraint equations: the momentum constraint vanishes for this TT, constant-trace ansatz, and the Hamiltonian constraint is equivalent to the displayed Lichnerowicz equation.

Exact positive constant solution of the CMC constraint equation: when $\tilde{R}=0$ and $\mathrm{\mid }\tilde{\sigma }{\mathrm{\mid }}_{\tilde{h}}^2=A>0$ is constant, the unique positive constant solution for $\tau \mathrm{\ne }0$ is $(3A\mathrm{/}(2{\tau }^2){)}^{1\mathrm{/}12}$.

Christoffel formula for the levi civita connection: in a chart, ${\mathrm{\Gamma }}^k{}_{ij}=\frac{1}{2}{\tilde{h}}^{k\mathrm{\ell }}({\mathrm{\partial }}_i{\tilde{h}}_{j\mathrm{\ell }}+{\mathrm{\partial }}_j{\tilde{h}}_{i\mathrm{\ell }}-{\mathrm{\partial }}_{\mathrm{\ell }}{\tilde{h}}_{ij})$.

Dual connection, Product connection on tensor and hom bundles: the covariant derivative on covariant tensor fields obeys the dual and product Leibniz rules.

Curvature is a type (1,3) tensor: curvature is tensorial, so its components in a coordinate basis determine it pointwise.

The 2 pi-periodic three-torus, its normalized measure, test functions and the Sobolev spaces H^s: ${\mathbb{T}}_{2\pi }^3=(\mathbb{R}\mathrm{/}2\pi \mathbb{Z}{)}^3$ has the quotient/product topology; smooth maps are those with smooth periodic lifts to ${\mathbb{R}}^3$. The angular expressions here are local coordinates and invariant differentials under translations by $2\pi {\mathbb{Z}}^3$.

Given: The flat three-torus with $L_{\ast }>0$, zero TT tensor and zero mean curvature parameter, together with the normalized measure and integer Sobolev spaces of the torus supplier. Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ for the Fourier Sobolev suppliers and inherited embedded Lorentzian interpretation.

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), Vacuum conformal constraint data, The 2 pi-periodic three-torus, its normalized measure, test functions and the Sobolev spaces H^s, Periodic Fourier differentiation identity and equivalence of the derivative and Fourier-weight Sobolev norms: ${\mathrm{A}\mathrm{C}}_{\omega }$ is the axiom that every countable family of nonempty sets has a choice function; the Fourier Sobolev suppliers state it explicitly, and the local GR definition carries it only for embedded interpretation.

Vacuum conformal constraint data: the Lichnerowicz equation uses $\tilde{R}$, $\tilde{\sigma }$, $\tau$ and $\tilde{\mathrm{\Delta }}={\mathrm{div}}_{\tilde{h}}{\mathrm{grad}}_{\tilde{h}}$.

Conformal reduction of the vacuum constraint equations: the vacuum constraint equations are equivalent to the scalar equation once the automatic momentum constraint is used.

The 2 pi-periodic three-torus, its normalized measure, test functions and the Sobolev spaces H^s: the torus has characters ${\chi }_k=e^{ik\cdot x}$, normalized measure $\mu$, and derivative Sobolev spaces $H^s$.

Periodic Fourier differentiation identity and equivalence of the derivative and Fourier-weight Sobolev norms: Fourier coefficients differentiate by multiplication by $ik$, and the Fourier-weight norm is equivalent to the derivative Sobolev norm.