The Yamabe quotient and its Euler–Lagrange equation

On a Riemannian manifold (M, g) of dimension n, the Yamabe quotient of a function u is

Q_g(u) = (∫ (aₙ |grad u|² + Scal_g u²) dV_g) / (∫ |u| ^ pₙ dV_g) ^ (2 / pₙ),

with aₙ = 4 (n - 1) / (n - 2) and pₙ = 2 n / (n - 2) (TauCeti.Yamabe.yamabeQuotient). It is invariant under multiplication of u by a nonzero constant. For a positive function u on a closed manifold, Q_g(u) is the volume-normalized total scalar curvature of the conformal metric u ^ (4 / (n - 2)) g, by TauCeti.Yamabe.integral_scalarCurvature_rescale_div_riemannianTotalVolume_rpow together with the energy identity TauCeti.Yamabe.integral_mul_conformalLaplacian_self, and the Yamabe problem asks for a minimizer.

This file computes the first variation of Q_g on a compact manifold. Write N(u) = ∫ |u| ^ pₙ dV. For C¹ functions u ≠ 0 and v,

d/dt Q_g(u + t v) |_(t = 0) = 2 ((∫ (aₙ ⟪grad v, grad u⟫ + Scal_g v u) dV) / N(u) ^ (2 / pₙ) - Q_g(u) (∫ |u| ^ (pₙ - 2) u v dV) / N(u)).

The energy is quadratic in t, and |·| ^ pₙ is differentiable with derivative pₙ |s| ^ (pₙ - 2) s because pₙ > 1, so N(u + t v) is differentiated under the integral sign. On a closed manifold, integration by parts turns the first term into ∫ v L_g u dV, where L_g is the conformal Laplacian. Since smooth functions separate continuous functions under integration, this gives the Euler–Lagrange equation: a nonzero C² function u is a critical point of Q_g in every smooth direction exactly when it solves

L_g u = λ |u| ^ (pₙ - 2) u

for some constant λ. Integrating the equation against u identifies the constant: Q_g(u) = λ N(u) ^ (1 - 2 / pₙ), so λ = Q_g(u) when u has unit L^pₙ norm. For a nonnegative u the equation is the Yamabe equation L_g u = λ u ^ (pₙ - 1); for a positive smooth u it says that u ^ (4 / (n - 2)) g has constant scalar curvature λ (TauCeti.Yamabe.scalarCurvature_rescale_eq_iff_conformalLaplacian_eq).

The quotient is defined for every function, through the Riemannian gradient and Bochner integrals; its classical meaning and variational results concern differentiable functions, while scale invariance (yamabeQuotient_smul) and nonnegativity under nonnegative scalar curvature hold for every function. The metric is the RiemannianBundle instance of the tangent bundle and the volume is TauCeti.riemannianVolume, as for TauCeti.Yamabe.conformalLaplacian.

Main declarations

All in the namespace TauCeti.Yamabe.

• yamabeQuotient: the Yamabe quotient Q_g(u).
• yamabeQuotient_nonneg_of_scalarCurvature_nonneg: nonnegative scalar curvature gives nonnegative quotients, without regularity assumptions on u.
• yamabeQuotient_smul: Q_g(c u) = Q_g(u) for c ≠ 0.
• yamabeQuotient_one: on a compact manifold, Q_g(1) = (∫ Scal_g dV_g) / Vol(g) ^ (2 / pₙ).
• hasDerivAt_yamabeQuotient_add_smul: the first variation of Q_g.
• forall_hasDerivAt_yamabeQuotient_add_smul_iff_exists_conformalLaplacian_eq: the Euler–Lagrange equation L_g u = λ |u| ^ (pₙ - 2) u characterizes the critical points of Q_g.
• forall_hasDerivAt_yamabeQuotient_add_smul_iff_exists_conformalLaplacian_eq_of_nonneg: for u ≥ 0 it is the Yamabe equation L_g u = λ u ^ (pₙ - 1).
• yamabeQuotient_eq_of_conformalLaplacian_eq: a solution of the Euler–Lagrange equation has Q_g(u) = λ N(u) ^ (1 - 2 / pₙ), and yamabeQuotient_eq_of_conformalLaplacian_eq_of_integral_eq_one: Q_g(u) = λ at unit norm.

References

• J. M. Lee and T. H. Parker, The Yamabe problem, Bull. Amer. Math. Soc. 17 (1987), §2 (the Yamabe functional, its Euler–Lagrange equation and the identification of the multiplier).

noncomputable def TauCeti.Yamabe.yamabeQuotient {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] (u : M → ℝ) : ℝ

The Yamabe quotient Q_g(u) = (∫ (aₙ |grad u|² + Scal_g u²) dV_g) / (∫ |u| ^ pₙ dV_g) ^ (2 / pₙ) of a function u on a Riemannian manifold of dimension n, where aₙ = 4 (n - 1) / (n - 2) and pₙ = 2 n / (n - 2). Its classical meaning is for differentiable u.

theorem TauCeti.Yamabe.yamabeQuotient_def {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] (u : M → ℝ) : yamabeQuotient I u = (∫ (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) / (∫ (x : M), |u x| ^ criticalExponent (Module.finrank ℝ E) ∂riemannianVolume I M) ^ (2 / criticalExponent (Module.finrank ℝ E))

Unfolding lemma for TauCeti.Yamabe.yamabeQuotient.

theorem TauCeti.Yamabe.yamabeQuotient_nonneg_of_scalarCurvature_nonneg {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] (hn : 2 < Module.finrank ℝ E) (u : M → ℝ) (hS : ∀ (x : M), 0 ≤ (CovariantDerivative.leviCivitaConnection I M).scalarCurvature x) : 0 ≤ yamabeQuotient I u

Nonnegative scalar curvature makes the total Yamabe quotient nonnegative in dimension n > 2, even without regularity or integrability assumptions on the function.

theorem TauCeti.Yamabe.yamabeQuotient_smul {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] (hn : 2 < Module.finrank ℝ E) {u : M → ℝ} {c : ℝ} (hc : c ≠ 0) : yamabeQuotient I (c • u) = yamabeQuotient I u

Scale invariance of the Yamabe quotient. In dimension n > 2, multiplying a function by a nonzero constant does not change its Yamabe quotient.

@[simp]

theorem TauCeti.Yamabe.yamabeQuotient_one {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] [CompactSpace M] : (yamabeQuotient I fun (x : M) => 1) = (∫ (x : M), (CovariantDerivative.leviCivitaConnection I M).scalarCurvature x ∂riemannianVolume I M) / riemannianTotalVolume I M ^ (2 / criticalExponent (Module.finrank ℝ E))

On a compact manifold of dimension n, the Yamabe quotient of the constant function 1 is the volume-normalized total scalar curvature (∫ Scal_g dV_g) / Vol(g) ^ (2 / pₙ).

The first variation

theorem TauCeti.Yamabe.hasDerivAt_yamabeQuotient_add_smul {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] [CompactSpace M] (hn : 2 < Module.finrank ℝ E) {u v : M → ℝ} (hu : ContMDiff I (modelWithCornersSelf ℝ ℝ) 1 u) (hv : ContMDiff I (modelWithCornersSelf ℝ ℝ) 1 v) (hu0 : u ≠ 0) : HasDerivAt (fun (t : ℝ) => yamabeQuotient I (u + t • v)) (2 * ((∫ (x : M), conformalCoefficient (Module.finrank ℝ E) * inner ℝ (Manifold.mgradient I v x) (Manifold.mgradient I u x) + (CovariantDerivative.leviCivitaConnection I M).scalarCurvature x * (v x * u x) ∂riemannianVolume I M) / (∫ (x : M), |u x| ^ criticalExponent (Module.finrank ℝ E) ∂riemannianVolume I M) ^ (2 / criticalExponent (Module.finrank ℝ E)) - (yamabeQuotient I u * ∫ (x : M), |u x| ^ (criticalExponent (Module.finrank ℝ E) - 2) * u x * v x ∂riemannianVolume I M) / ∫ (x : M), |u x| ^ criticalExponent (Module.finrank ℝ E) ∂riemannianVolume I M)) 0

The first variation of the Yamabe quotient. On a compact manifold of dimension n > 2, let u ≠ 0 and v be C¹ functions, and write N(u) = ∫ |u| ^ pₙ dV. Then t ↦ Q_g(u + t v) has derivative 2 ((∫ (aₙ ⟪grad v, grad u⟫ + Scal_g v u) dV) / N(u) ^ (2 / pₙ) - Q_g(u) (∫ |u| ^ (pₙ - 2) u v dV) / N(u)) at t = 0.

theorem TauCeti.Yamabe.yamabeQuotient_eq_of_conformalLaplacian_eq {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] [CompactSpace M] [I.Boundaryless] (hn : 2 < Module.finrank ℝ E) {u : M → ℝ} (hu : ContMDiff I (modelWithCornersSelf ℝ ℝ) 2 u) {c : ℝ} (h : ∀ (x : M), conformalLaplacian I u x = c * (|u x| ^ (criticalExponent (Module.finrank ℝ E) - 2) * u x)) : yamabeQuotient I u = c * (∫ (x : M), |u x| ^ criticalExponent (Module.finrank ℝ E) ∂riemannianVolume I M) ^ (1 - 2 / criticalExponent (Module.finrank ℝ E))

The multiplier of the Euler–Lagrange equation. On a closed manifold of dimension n > 2, if a C² function u solves L_g u = λ |u| ^ (pₙ - 2) u, then Q_g(u) = λ (∫ |u| ^ pₙ dV) ^ (1 - 2 / pₙ).

theorem TauCeti.Yamabe.yamabeQuotient_eq_of_conformalLaplacian_eq_of_integral_eq_one {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] [CompactSpace M] [I.Boundaryless] (hn : 2 < Module.finrank ℝ E) {u : M → ℝ} (hu : ContMDiff I (modelWithCornersSelf ℝ ℝ) 2 u) {c : ℝ} (h : ∀ (x : M), conformalLaplacian I u x = c * (|u x| ^ (criticalExponent (Module.finrank ℝ E) - 2) * u x)) (h1 : ∫ (x : M), |u x| ^ criticalExponent (Module.finrank ℝ E) ∂riemannianVolume I M = 1) : yamabeQuotient I u = c

The multiplier at unit norm. On a closed manifold of dimension n > 2, if a C² function u with ∫ |u| ^ pₙ dV = 1 solves L_g u = λ |u| ^ (pₙ - 2) u, then λ = Q_g(u).

theorem TauCeti.Yamabe.forall_hasDerivAt_yamabeQuotient_add_smul_iff_exists_conformalLaplacian_eq {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] [CompactSpace M] [I.Boundaryless] (hn : 2 < Module.finrank ℝ E) {u : M → ℝ} (hu : ContMDiff I (modelWithCornersSelf ℝ ℝ) 2 u) (hu0 : u ≠ 0) : (∀ (v : M → ℝ), ContMDiff I (modelWithCornersSelf ℝ ℝ) (↑⊤) v → HasDerivAt (fun (t : ℝ) => yamabeQuotient I (u + t • v)) 0 0) ↔ ∃ (c : ℝ), ∀ (x : M), conformalLaplacian I u x = c * (|u x| ^ (criticalExponent (Module.finrank ℝ E) - 2) * u x)

The Euler–Lagrange equation of the Yamabe quotient. On a closed manifold of dimension n > 2, a nonzero C² function u is a critical point of the Yamabe quotient, in the sense that t ↦ Q_g(u + t v) has derivative 0 at t = 0 for every smooth v, exactly when it solves L_g u = λ |u| ^ (pₙ - 2) u for some constant λ.

theorem TauCeti.Yamabe.forall_hasDerivAt_yamabeQuotient_add_smul_iff_exists_conformalLaplacian_eq_of_nonneg {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] [CompactSpace M] [I.Boundaryless] (hn : 2 < Module.finrank ℝ E) {u : M → ℝ} (hu : ContMDiff I (modelWithCornersSelf ℝ ℝ) 2 u) (hu0 : u ≠ 0) (hnn : 0 ≤ u) : (∀ (v : M → ℝ), ContMDiff I (modelWithCornersSelf ℝ ℝ) (↑⊤) v → HasDerivAt (fun (t : ℝ) => yamabeQuotient I (u + t • v)) 0 0) ↔ ∃ (c : ℝ), ∀ (x : M), conformalLaplacian I u x = c * u x ^ (criticalExponent (Module.finrank ℝ E) - 1)

The Euler–Lagrange equation for nonnegative functions. On a closed manifold of dimension n > 2, a nonzero nonnegative C² function u is a critical point of the Yamabe quotient exactly when it solves the Yamabe equation L_g u = λ u ^ (pₙ - 1) for some constant λ.