The energy of the conformal Laplacian

On a Riemannian manifold (M, g) without boundary of dimension n, Green's first identity ∫ f Δu dV = -∫ ⟪grad f, grad u⟫ dV turns the conformal Laplacian L_g u = aₙ Δ⁺_g u + Scal_g u, with aₙ = 4 (n - 1) / (n - 2) and Δ⁺ = -Δ, into its energy form:

∫ f L_g u dV_g = ∫ (aₙ ⟪grad f, grad u⟫ + Scal_g f u) dV_g

for a compactly supported C¹ function f and a C² function u. The right-hand side is symmetric in f and u, so L_g is formally self-adjoint, and for f = u it is the energy

∫ u L_g u dV_g = ∫ (aₙ |grad u|² + Scal_g u²) dV_g,

the numerator of the Yamabe quotient Q_g(u). On a closed manifold every function has compact support, so these identities hold for all C² functions.

Combined with the conformal covariance L_ĝ v = u ^ (-(n + 2) / (n - 2)) L_g (u v) of the conformal Laplacian, whose integrated form is TauCeti.Yamabe.integral_mul_conformalLaplacian_rescale, the energy identity shows that the energy is conformally invariant: for ĝ = u ^ (4 / (n - 2)) g with u positive and every compactly supported C² function v,

∫ (aₙ |grad_ĝ v|² + Scal_ĝ v²) dV_ĝ = ∫ (aₙ |grad_g (u v)|² + Scal_g (u v)²) dV_g.

Together with the invariance ∫ |v| ^ pₙ dV_ĝ = ∫ |u v| ^ pₙ dV_g of the denominator (TauCeti.WithMetric.integral_abs_rpow_rescale_rpow), this is the invariance Q_ĝ(v) = Q_g(u v) of the Yamabe quotient on compactly supported C² functions.

As in TauCeti.Yamabe.integral_mul_conformalLaplacian_rescale, the conformal invariance compares the volumes of the metric-indexed copies TauCeti.WithMetric g and TauCeti.WithMetric ĝ of M, while the gradient and scalar curvature of each metric are taken with that metric installed as the RiemannianBundle instance of M. The volume of a copy is transported to M by TauCeti.WithMetric.integral_comp_ofWithMetric.

Main results

• TauCeti.Yamabe.integral_mul_conformalLaplacian: ∫ f L_g u dV = ∫ (aₙ ⟪grad f, grad u⟫ + Scal_g f u) dV.
• TauCeti.Yamabe.integral_mul_conformalLaplacian_right_eq_left: the formal self-adjointness ∫ f L_g u dV = ∫ (L_g f) u dV.
• TauCeti.Yamabe.integral_mul_conformalLaplacian_self: the energy identity ∫ u L_g u dV = ∫ (aₙ |grad u|² + Scal_g u²) dV.
• TauCeti.Yamabe.integral_energy_rescale: the conformal invariance of the energy.

References

• J. M. Lee and T. H. Parker, The Yamabe problem, Bull. Amer. Math. Soc. 17 (1987), §2 (the energy form of the Yamabe functional and its conformal invariance).

theorem TauCeti.Yamabe.integral_mul_conformalLaplacian {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {H : Type u_2} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type u_3} [TopologicalSpace M] [ChartedSpace H M] [T2Space M] [SigmaCompactSpace M] [IsManifold I (↑⊤) M] [MeasurableSpace M] [BorelSpace M] [Bundle.RiemannianBundle fun (x : M) => TangentSpace I x] [IsContMDiffRiemannianBundle I (↑⊤) E fun (x : M) => TangentSpace I x] [IsContinuousRiemannianBundle E fun (x : M) => TangentSpace I x] [I.Boundaryless] {f u : M → ℝ} (hf : ContMDiff I (modelWithCornersSelf ℝ ℝ) 1 f) (hfc : HasCompactSupport f) (hu : ContMDiff I (modelWithCornersSelf ℝ ℝ) 2 u) : ∫ (x : M), f x * conformalLaplacian I u x ∂riemannianVolume I M = ∫ (x : M), conformalCoefficient (Module.finrank ℝ E) * inner ℝ (Manifold.mgradient I f x) (Manifold.mgradient I u x) + (CovariantDerivative.leviCivitaConnection I M).scalarCurvature x * (f x * u x) ∂riemannianVolume I M

Integration by parts for the conformal Laplacian on a manifold without boundary: for a compactly supported C¹ function f and a C² function u, ∫ f L_g u dV = ∫ (aₙ ⟪grad f, grad u⟫ + Scal_g f u) dV, where L_g is the conformal Laplacian and aₙ = 4 (n - 1) / (n - 2).

theorem TauCeti.Yamabe.integral_mul_conformalLaplacian_right_eq_left {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {H : Type u_2} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type u_3} [TopologicalSpace M] [ChartedSpace H M] [T2Space M] [SigmaCompactSpace M] [IsManifold I (↑⊤) M] [MeasurableSpace M] [BorelSpace M] [Bundle.RiemannianBundle fun (x : M) => TangentSpace I x] [IsContMDiffRiemannianBundle I (↑⊤) E fun (x : M) => TangentSpace I x] [IsContinuousRiemannianBundle E fun (x : M) => TangentSpace I x] [I.Boundaryless] {f u : M → ℝ} (hf : ContMDiff I (modelWithCornersSelf ℝ ℝ) 2 f) (hfc : HasCompactSupport f) (hu : ContMDiff I (modelWithCornersSelf ℝ ℝ) 2 u) : ∫ (x : M), f x * conformalLaplacian I u x ∂riemannianVolume I M = ∫ (x : M), conformalLaplacian I f x * u x ∂riemannianVolume I M

The conformal Laplacian is formally self-adjoint on a manifold without boundary: for a compactly supported C² function f and a C² function u, ∫ f L_g u dV = ∫ (L_g f) u dV.

theorem TauCeti.Yamabe.integral_mul_conformalLaplacian_self {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {H : Type u_2} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type u_3} [TopologicalSpace M] [ChartedSpace H M] [T2Space M] [SigmaCompactSpace M] [IsManifold I (↑⊤) M] [MeasurableSpace M] [BorelSpace M] [Bundle.RiemannianBundle fun (x : M) => TangentSpace I x] [IsContMDiffRiemannianBundle I (↑⊤) E fun (x : M) => TangentSpace I x] [IsContinuousRiemannianBundle E fun (x : M) => TangentSpace I x] [I.Boundaryless] {u : M → ℝ} (hu : ContMDiff I (modelWithCornersSelf ℝ ℝ) 2 u) (huc : HasCompactSupport u) : ∫ (x : M), u x * conformalLaplacian I u x ∂riemannianVolume I M = ∫ (x : M), conformalCoefficient (Module.finrank ℝ E) * ‖Manifold.mgradient I u x‖ ^ 2 + (CovariantDerivative.leviCivitaConnection I M).scalarCurvature x * u x ^ 2 ∂riemannianVolume I M

The energy identity for the conformal Laplacian on a manifold without boundary: for a compactly supported C² function u, ∫ u L_g u dV = ∫ (aₙ |grad u|² + Scal_g u²) dV.

theorem TauCeti.Yamabe.integral_energy_rescale {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {H : Type u_2} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} [I.Boundaryless] {M : Type u_3} [TopologicalSpace M] [ChartedSpace H M] [T2Space M] [SigmaCompactSpace M] [IsManifold I (↑⊤) M] (g : Bundle.ContMDiffRiemannianMetric I (↑⊤) E fun (x : M) => TangentSpace I x) (hn : Module.finrank ℝ E ≠ 2) {u : M → ℝ} (hu_pos : ∀ (x : M), 0 < u x) {hf : ContMDiff I (modelWithCornersSelf ℝ ℝ) ↑⊤ fun (x : M) => u x ^ (4 / (↑(Module.finrank ℝ E) - 2))} {hf_pos : ∀ (x : M), 0 < u x ^ (4 / (↑(Module.finrank ℝ E) - 2))} {v : M → ℝ} (hv : ContMDiff I (modelWithCornersSelf ℝ ℝ) 2 v) (hvc : HasCompactSupport v) : ∫ (x : WithMetric (g.rescale (fun (x : M) => u x ^ (4 / (↑(Module.finrank ℝ E) - 2))) hf hf_pos)), conformalCoefficient (Module.finrank ℝ E) * ‖Manifold.mgradient I v x.ofWithMetric‖ ^ 2 + (g.rescale (fun (x : M) => u x ^ (4 / (↑(Module.finrank ℝ E) - 2))) hf hf_pos).scalarCurvature x.ofWithMetric * v x.ofWithMetric ^ 2 ∂riemannianVolume I (WithMetric (g.rescale (fun (x : M) => u x ^ (4 / (↑(Module.finrank ℝ E) - 2))) hf hf_pos)) = ∫ (x : WithMetric g), conformalCoefficient (Module.finrank ℝ E) * ‖Manifold.mgradient I (u * v) x.ofWithMetric‖ ^ 2 + g.scalarCurvature x.ofWithMetric * (u * v) x.ofWithMetric ^ 2 ∂riemannianVolume I (WithMetric g)

Conformal invariance of the Yamabe energy. In dimension n ≠ 2, let u be a positive function such that ĝ = u ^ (4 / (n - 2)) g is a smooth metric on a manifold without boundary. Then for every compactly supported C² function v, ∫ (aₙ |grad_ĝ v|² + Scal_ĝ v²) dV_ĝ = ∫ (aₙ |grad_g (u v)|² + Scal_g (u v)²) dV_g, where aₙ = 4 (n - 1) / (n - 2).