0.0.1 Constraints. LCBY equations

Thin sanwich formulation. Given:

riemannian manifold (M, γ), scalar τ. Traceless tensor u^{ij}, scalar N. Sources, fields and matter: 4 scalars ρ_{I} ≥ 0, 2 vectors J_{1}, J_{2}. Unknowns: scalar φ, vector β Equations

H ≡ Δ_{γ}φ − f(., β, φ) = 0,

f(., β, φ) ≡ r̃φ − aφ^{−P} − q_{1}φ^{−P_{1}} + (b − q_{2})φ^{Q},

P= \frac{3n−2}{n−2}, P_{1} = \frac{n}{n−2}, Q = \frac{n+2}{n−2}, k_{n} = \frac{n}{4n−1}

r̃ ≡ r − q_{3}, r ≡ k_{n}R(γ), a ≡ k_{n}A.A + q_{0} ≥ 0

b ≡ \frac{n − 2}{4n}τ^{2} ≥ 0, q_{I} ≡ 2k_{n}ρ_{I} ≥ 0.

A^{ij} ≡ (2N)^{−1}{(L_{γ,conf}β)^{ij} − u^{ij}}

M^{i} ≡ D_{j}A^{ij} − \{\frac{n − 1}{n}φ^{2n/(n−2)}∂^{i}τ + φ^{2(n+2)/(n−2)}J_{2} + J_{1}\} = 0.

Note: ψ scalar field (source)

2ρ_{0} ≡ N^{−2}|∂_{0}ψ|^{2}, 2ρ_{3} ≡ γ^{ij}∂_{i}ψ∂_{j}ψ

0.0.2 Momentum constraint

Suppose momentarily φ is known if ∂τ ≢ 0 or J_{2} ≢ 0. Linear elliptic for β . Theorem. (as. euc. case) (M, γ) : W_{σ,ρ}^{p} as. euc., N > 0, N − 1 ∈ W_{σ,ρ}^{p} , σ > \frac{n}{p} + 1, ρ > −\frac{n}{p}, p > \frac{n}{2}. u, τ ∈ W_{s_{0}+1,δ+1}^{p} , J_{1}, J_{2} ∈ W_{s_{0},δ+2}^{p}, s_{0} ≥ 0, φ > 0, (1 − φ) ∈ W_{s_{0}+2,δ}^{p} . Then the momentum constraint has one and only one solution β ∈ W_{s+2,δ}^{p}

with 0 ≤ s ≤ Max(s_{0}, σ − 2), −\frac{n}{p} < δ < n − 2 − \frac{n}{p}

Case of compact M, same result without the weights, if (M, γ) has no conformal Killing vector, or, when it has one, if the given data and sources are invariant under it.

0.0.3 Theorem. (M compact)

The (L) equation in vacuum on (M, γ), γ ∈ W_{σ}^{p}, σ ≥ \frac{n}{p} + 1, a, b ∈ L^{p} ∩ L^{∞}, p > \frac{n}{2} , admits a solution φ > 0, φ ∈ W_{2}^{p} if:

1. (M, γ) is in the positive Yamabe class, r(γ) = 1 and a ≢ 0 on M.

2. (M, γ) is in the zero Yamabe class, r(γ)= 0, τ^{2} ≢ 0 and a ≢ 0 on M. 3.(M, γ) is in the negative Yamabe class, r(γ) = −1 and Inf τ^{2} > 0 on a

sufficiently large subset of M.

Proof: super and subsolution method. Local conformal transformations when necessary.

Case of coupling with a scalar field: ã = a + q_{0}, r̃ = r − q_{3}, b̃ = b − mψ^{2} 0.0.4 Definition

(M, γ) is (p, σ, ρ) as. euc. if γ − e ∈ W_{σ,ρ}^{p} , σ > \frac{n}{p} + 1, ρ > −\frac{n}{p}, then γ − e ∈ C_{α}^{1}, α > 0.

0.0.5 Theorem.

Let (M, γ) be (p, σ, ρ) as. euc. i.e. γ − e ∈ W_{σ,ρ}^{p}, σ > \frac{n}{p} + 1, ρ > −\frac{n}{p}.

The operator Δ_{γ} − k, k ∈ W_{s_{0},δ_{0}+2}^{p}, s_{0} ≥ 0, δ_{0} > −\frac{n}{p}, is an isomorphism from W_{s+2,δ}^{p} onto W_{s,δ+2}^{p}, with 0 ≤ s ≤ Max(σ − 1, s_{0}), if

p > \frac{n}{2}, −\frac{n}{p} < δ < −\frac{n}{p} + n − 2.

and if

∫_{M}{|∂f|^{2} + kf^{2}}μ_{γ} > 0 for all f ∈ D, f ≢ 0 .

Proof. Δ_{γ} − k, k ∈ W_{0,δ_{0}+2}^{p}, δ_{0} > −\frac{n}{p}, is injective on W_{2,δ}^{p} if p > \frac{n}{2}, δ > −\frac{n}{p}, because if u ∈ ker(Δ_{γ} − k) ∩ W_{2,δ}^{p}, δ > −\frac{n}{p}, then u ∈ W_{2,δ}^{p} for any δ < −2 + n − \frac{n}{p}, hence for some δ > −1 + \frac{n}{2} − \frac{n}{p}.

Integration after product by u and Holder inequality complete the proof.

0.0.6 Maximum principle.

If c − θ ∈ W_{2,δ}^{p}, p > \frac{n}{2}, δ > −\frac{n}{p}, with c a number ≥ 0 and if θ satisfies the equation

Δ_{γ}θ − kθ = −f

with γ − e ∈ W_{σ,ρ}^{p}, σ > \frac{n}{p} + 1, ρ > −\frac{n}{p}, k ∈ W_{0,δ+2}^{p}, k ≥ 0 , and f ≥ 0, then θ ≥ 0 on M.

Proof. The lemma holds by the classical maximum principle when c − θ ∈ W_{s+2,δ}^{p} if s > \frac{n}{p}, δ > −\frac{n}{p}, and k is bounded, since then θ ∈ C^{2} and tends to c at infinity. Approximate f and k in W_{0,δ+2}^{p} by sequences f_{n}, k_{n} ∈ W_{s,δ+2}^{p}, s > \frac{n}{p}, f_{n} ≥ 0, k_{n} ≥ 0 and use the isomorphism theorem.

Theorem (Lichnerowicz equation, scaled sources, n=3). Let (M, γ) be a (p, σ, ρ) asymptotically euclidean 3- manifold (M, γ), σ >

\frac{3}{p} + 1, ρ > −\frac{3}{p}, p > \frac{3}{2} in the positive Yamabe class, conformally transformed so that r(γ) = 0. The equation

Δ_{γ}φ + aφ^{−7} + qφ^{−3} − bφ^{5} = 0,

a, q, b ∈ W_{0,δ+2}^{p}, δ > −\frac{3}{p}, a, q, b ≥ 0,

has one and only one solution, φ = 1 + u, u ∈ W_{2,δ}^{p}, φ > 0 >, if δ < 1 − \frac{3}{p}. It can be obtained by iteration.

If moreover a, q, b ∈ W_{s_{0},δ+2}^{p}, s_{0} ≥ 0, then u ∈ W_{s+2,δ}^{p}, s = Min(s_{0}, σ − 1).

0.0.7 Unscaled sources.

0.0.8 Theorem (n=3)

Let a, d ≡ b−q_{2} ∈ W_{s_{0},δ+2}^{p} a, d ≥ 0, s_{0} ≥ 0, δ > −\frac{3}{p}, p > \frac{3}{2}, be given on (M, γ), (p, σ, ρ) as. euc. positive Yamabe. The (L) equation

△_{γ}φ − r(γ)φ + aφ^{−7} + dφ^{5} = 0

has a solution φ > 0, with 1 − φ ∈ W_{s+2,δ}^{p}, s = Inf(s_{0}, σ − 1), −\frac{3}{p} < δ < 1 − \frac{3}{p} if on M it holds that either

• 1. d + a ≤ r , infz_{1}(x) > 0, and inf_{x∈M} z_{2}(x) ≥ max {1, sup_{x∈M} z_{1}(x)} , where z_{1}(x) and z_{2}(x) are the two positive roots of the polynomial

P_{x}(z) ≡ d(x)z^{3} − r(x)z^{2} + a(x).

• or 2.

||d||_{W_{0,δ+2}^{p}} ≤ α, α = C\{\frac{1}{1 + ||a||_{W_{0,δ+2}^{p}}} \}^{1/4}

C > 0 depends only on (M, γ).

Proof. 1. Super - subsolutions.

2. Iterate, starting from u_{0} = 1 − φ_{0} = 0. Then u_{n} ≥ 0. Prove uniform bounds in W_{2,δ}^{p}