Total scalar curvature of a conformal metric

Let g be a smooth Riemannian metric on a manifold M of dimension n ≠ 2, and let ĝ = u ^ (4 / (n - 2)) g for a positive function u. The scalar curvature of ĝ is Scal_ĝ = u ^ (-(n + 2) / (n - 2)) L_g u, where L_g is the conformal Laplacian, and the volume of ĝ has density u ^ pₙ with respect to that of g, where pₙ = 2 n / (n - 2). Since pₙ - (n + 2) / (n - 2) = 1, the two factors combine into the identity of densities

Scal_ĝ dV_ĝ = u L_g u dV_g.

Integrating it gives the total scalar curvature ∫ Scal_ĝ dV_ĝ = ∫ u L_g u dV_g, and dividing by Vol(ĝ) ^ (2 / pₙ) = (∫ u ^ pₙ dV_g) ^ (2 / pₙ) gives the identity between the volume-normalized total scalar curvature of ĝ and a quotient of integrals against g:

(∫ Scal_ĝ dV_ĝ) / Vol(ĝ) ^ (2 / pₙ) = (∫ u L_g u dV_g) / (∫ u ^ pₙ dV_g) ^ (2 / pₙ).

On a closed manifold, integration by parts turns ∫ u L_g u dV_g into the energy ∫ (aₙ |du|² + Scal_g u²) dV_g, so the right-hand side is the Yamabe quotient Q_g(u); this file does not perform that integration by parts. When u is a positive solution of the Yamabe equation L_g u = λ u ^ (pₙ - 1), the numerator is ∫ u L_g u dV_g = λ ∫ u ^ pₙ dV_g, so the quotient equals λ when u has unit L^pₙ norm.

The same density computation applies to the conformal covariance L_ĝ v = u ^ (-(n + 2) / (n - 2)) L_g (u v) of the conformal Laplacian: for every C² function v, v L_ĝ v dV_ĝ = (u v) L_g (u v) dV_g, so ∫ v L_ĝ v dV_ĝ = ∫ (u v) L_g (u v) dV_g. Together with TauCeti.WithMetric.integral_abs_rpow_rescale_rpow for the denominator, this is the invariance of the quotient (∫ v L v dV) / (∫ |v| ^ pₙ dV) ^ (2 / pₙ) under (ĝ, v) ↦ (g, u v) for C² functions, before integration by parts.

The integrals are Bochner integrals and the identities need no integrability hypothesis: the two integrands agree pointwise once the volume of ĝ is written as a density against that of g. As in TauCeti.WithMetric.integral_riemannianVolume_rescale, the two volumes live on the metric-indexed copies TauCeti.WithMetric g and TauCeti.WithMetric ĝ of M, while the conformal Laplacian of g is taken with g installed as the RiemannianBundle instance of M.

Main results

• TauCeti.Yamabe.rpow_criticalExponent_mul_scalarCurvature_rescale: the pointwise identity u ^ pₙ Scal_ĝ = u L_g u.
• TauCeti.Yamabe.integral_scalarCurvature_rescale: ∫ Scal_ĝ dV_ĝ = ∫ u L_g u dV_g.
• TauCeti.Yamabe.integral_scalarCurvature_rescale_div_riemannianTotalVolume_rpow: on a compact manifold, (∫ Scal_ĝ dV_ĝ) / Vol(ĝ) ^ (2 / pₙ) = (∫ u L_g u dV_g) / (∫ u ^ pₙ dV_g) ^ (2 / pₙ).
• TauCeti.Yamabe.integral_mul_conformalLaplacian_rescale: ∫ v L_ĝ v dV_ĝ = ∫ (u v) L_g (u v) dV_g.
• TauCeti.Yamabe.integral_mul_conformalLaplacian_eq_mul_integral_rpow: a positive solution of L_g u = c u ^ (pₙ - 1) has ∫ u L_g u dV_g = c ∫ u ^ pₙ dV_g.

References

• J. M. Lee and T. H. Parker, The Yamabe problem, Bull. Amer. Math. Soc. 17 (1987), §2.

theorem TauCeti.Yamabe.rpow_criticalExponent_mul_scalarCurvature_rescale {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] [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))} (x : M) : u x ^ criticalExponent (Module.finrank ℝ E) * (g.rescale (fun (x : M) => u x ^ (4 / (↑(Module.finrank ℝ E) - 2))) hf hf_pos).scalarCurvature x = u x * conformalLaplacian I u x

Scalar curvature density of a conformal metric. In dimension n ≠ 2, let u be a positive function such that ĝ = u ^ (4 / (n - 2)) g is a smooth metric. Then u ^ pₙ Scal_ĝ = u L_g u, where pₙ = 2 n / (n - 2) and L_g is the conformal Laplacian of g. Since u ^ pₙ is the density of the volume of ĝ against that of g, this is the identity of densities Scal_ĝ dV_ĝ = u L_g u dV_g.

theorem TauCeti.Yamabe.integral_mul_conformalLaplacian_rescale {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] [IsManifold I (↑⊤) M] [LindelofSpace 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) : ∫ (x : WithMetric (g.rescale (fun (x : M) => u x ^ (4 / (↑(Module.finrank ℝ E) - 2))) hf hf_pos)), v x.ofWithMetric * conformalLaplacian I v x.ofWithMetric ∂riemannianVolume I (WithMetric (g.rescale (fun (x : M) => u x ^ (4 / (↑(Module.finrank ℝ E) - 2))) hf hf_pos)) = ∫ (x : WithMetric g), (u * v) x.ofWithMetric * conformalLaplacian I (u * v) x.ofWithMetric ∂riemannianVolume I (WithMetric g)

The conformal Laplacian energy of a conformal metric. In dimension n ≠ 2, let u be a positive function such that ĝ = u ^ (4 / (n - 2)) g is a smooth metric. Then for every C² function v, ∫ v L_ĝ v dV_ĝ = ∫ (u v) L_g (u v) dV_g, where L_ĝ and L_g are the conformal Laplacians. No integrability hypothesis is needed: the two integrands agree once dV_ĝ is written as a density against dV_g.

theorem TauCeti.Yamabe.integral_scalarCurvature_rescale {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] [IsManifold I (↑⊤) M] [LindelofSpace 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))} : ∫ (x : WithMetric (g.rescale (fun (x : M) => u x ^ (4 / (↑(Module.finrank ℝ E) - 2))) hf hf_pos)), (g.rescale (fun (x : M) => u x ^ (4 / (↑(Module.finrank ℝ E) - 2))) hf hf_pos).scalarCurvature x.ofWithMetric ∂riemannianVolume I (WithMetric (g.rescale (fun (x : M) => u x ^ (4 / (↑(Module.finrank ℝ E) - 2))) hf hf_pos)) = ∫ (x : WithMetric g), u x.ofWithMetric * conformalLaplacian I u x.ofWithMetric ∂riemannianVolume I (WithMetric g)

Total scalar curvature of a conformal metric. In dimension n ≠ 2, let u be a positive function such that ĝ = u ^ (4 / (n - 2)) g is a smooth metric. Then ∫ Scal_ĝ dV_ĝ = ∫ u L_g u dV_g, where L_g is the conformal Laplacian of g. No integrability hypothesis is needed: the two integrands agree once dV_ĝ is written as a density against dV_g.

theorem TauCeti.Yamabe.integral_scalarCurvature_rescale_div_riemannianTotalVolume_rpow {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] [IsManifold I (↑⊤) M] [LindelofSpace M] [CompactSpace 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))} : (∫ (x : WithMetric (g.rescale (fun (x : M) => u x ^ (4 / (↑(Module.finrank ℝ E) - 2))) hf hf_pos)), (g.rescale (fun (x : M) => u x ^ (4 / (↑(Module.finrank ℝ E) - 2))) hf hf_pos).scalarCurvature x.ofWithMetric ∂riemannianVolume I (WithMetric (g.rescale (fun (x : M) => u x ^ (4 / (↑(Module.finrank ℝ E) - 2))) hf hf_pos))) / riemannianTotalVolume I (WithMetric (g.rescale (fun (x : M) => u x ^ (4 / (↑(Module.finrank ℝ E) - 2))) hf hf_pos)) ^ (2 / criticalExponent (Module.finrank ℝ E)) = (∫ (x : WithMetric g), u x.ofWithMetric * conformalLaplacian I u x.ofWithMetric ∂riemannianVolume I (WithMetric g)) / (∫ (x : WithMetric g), u x.ofWithMetric ^ criticalExponent (Module.finrank ℝ E) ∂riemannianVolume I (WithMetric g)) ^ (2 / criticalExponent (Module.finrank ℝ E))

Volume-normalized total scalar curvature of a conformal metric. On a compact manifold of dimension n ≠ 2, let u be a positive function such that ĝ = u ^ (4 / (n - 2)) g is a smooth metric, and let pₙ = 2 n / (n - 2). Then (∫ Scal_ĝ dV_ĝ) / Vol(ĝ) ^ (2 / pₙ) = (∫ u L_g u dV_g) / (∫ u ^ pₙ dV_g) ^ (2 / pₙ), where L_g is the conformal Laplacian of g.

theorem TauCeti.Yamabe.integral_mul_conformalLaplacian_eq_mul_integral_rpow {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] [IsManifold I (↑⊤) M] [LindelofSpace M] (g : Bundle.ContMDiffRiemannianMetric I (↑⊤) E fun (x : M) => TangentSpace I x) {u : M → ℝ} (hu_pos : ∀ (x : M), 0 < u x) {c : ℝ} (hu : ∀ (x : M), conformalLaplacian I u x = c * u x ^ (criticalExponent (Module.finrank ℝ E) - 1)) : ∫ (x : WithMetric g), u x.ofWithMetric * conformalLaplacian I u x.ofWithMetric ∂riemannianVolume I (WithMetric g) = c * ∫ (x : WithMetric g), u x.ofWithMetric ^ criticalExponent (Module.finrank ℝ E) ∂riemannianVolume I (WithMetric g)

Integral of the Yamabe equation. Let u be a positive solution of the Yamabe equation L_g u = c u ^ (pₙ - 1), where L_g is the conformal Laplacian and pₙ = 2 n / (n - 2). Then ∫ u L_g u dV_g = c ∫ u ^ pₙ dV_g.