Local solutions of the conformal flattening equation

Let g be a smooth Riemannian metric on a finite-dimensional real normed space E. In dimension at least 3, the metric e^(2φ) g has vanishing Schouten tensor exactly when φ solves the conformal flattening equation

Hess φ = P + dφ ⊗ dφ - (|dφ|² / 2) g,

with P the Schouten tensor of g (Bundle.ContMDiffRiemannianMetric.schoutenTensor_leviCivitaConnection_rescale_eq_zero_iff). This file proves that the equation is locally solvable with arbitrary initial data wherever the Weyl and Cotton tensors of g vanish: if W = 0 and C = 0 near x₀, then for every number c and tangent vector ξ at x₀ there is a smooth φ solving the equation near x₀ with φ x₀ = c and grad φ x₀ = ξ. This is a converse of TauCeti.Manifold.weylTensor_eq_zero_of_forall_exists_eventually_hessian_eq and TauCeti.Manifold.cottonTensor_eq_zero_of_eventually_hessian_eq, and the analytic step in the proof that a metric with vanishing Weyl tensor (dimension at least 4) or vanishing Cotton tensor (dimension 3) is locally conformally flat.

In linear coordinates, writing α = dφ, the equation is the first-order total differential equation D(φ, α)(u) = (α u, v ↦ α(Γ(u, v)) + Φ_α(u, v)) on E × (ℝ × E*), where Γ are the Christoffel symbols of g (TauCeti.Manifold.christoffelSecondKind) and Φ_α is the right side of the equation. Its Frobenius integrability condition is checked without computing curvature in coordinates: at a point (y, c, α), a quadratic function with this value, differential and prescribed second derivative solves the equation at y, and along such a function the antisymmetrized derivative of the defect of the equation is -(C(X, Y, Z) + W(Z, Y, X, grad φ)) (TauCeti.Manifold.mvfderiv_hessian_sub_conformalFlatteningForm_sub_swap), which vanishes. The local Frobenius theorem TauCeti.exists_contDiffOn_solution_family_of_isFrobeniusIntegrableAt then produces the solution, and a cutoff makes it globally smooth.

Main results

• TauCeti.Manifold.exists_eventually_hessian_eq_model_space: on a normed space with a smooth Riemannian metric whose Weyl and Cotton tensors vanish near x₀, the conformal flattening equation has a smooth solution near x₀ with any prescribed value and gradient at x₀.

References

• J. M. Lee and T. H. Parker, The Yamabe problem, Bull. Amer. Math. Soc. 17 (1987), Section 2.
• A. L. Besse, Einstein Manifolds, Springer (1987), Section 1.G (the Weyl–Schouten theorem).
• J. Dieudonné, Foundations of Modern Analysis, Academic Press (1969), Section 10.9 (the Frobenius theorem for total differential equations).

The conformal flattening system on a model space

theorem TauCeti.Manifold.exists_eventually_hessian_eq_model_space {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [Bundle.RiemannianBundle fun (x : E) => TangentSpace (modelWithCornersSelf ℝ E) x] [IsContMDiffRiemannianBundle (modelWithCornersSelf ℝ E) (↑⊤) E fun (x : E) => TangentSpace (modelWithCornersSelf ℝ E) x] {x₀ : E} (hW : ∀ᶠ (y : E) in nhds x₀, (CovariantDerivative.leviCivitaConnection (modelWithCornersSelf ℝ E) E).weylTensor y = 0) (hC : ∀ᶠ (y : E) in nhds x₀, (CovariantDerivative.leviCivitaConnection (modelWithCornersSelf ℝ E) E).cottonTensor y = 0) (c : ℝ) (ξ : TangentSpace (modelWithCornersSelf ℝ E) x₀) : ∃ (φ : E → ℝ), ContMDiff (modelWithCornersSelf ℝ E) (modelWithCornersSelf ℝ ℝ) (↑⊤) φ ∧ φ x₀ = c ∧ mgradient (modelWithCornersSelf ℝ E) φ x₀ = ξ ∧ ∀ᶠ (y : E) in nhds x₀, ∀ (u v : TangentSpace (modelWithCornersSelf ℝ E) y), hessian (modelWithCornersSelf ℝ E) φ y u v = (((CovariantDerivative.leviCivitaConnection (modelWithCornersSelf ℝ E) E).conformalFlatteningForm y (mgradient (modelWithCornersSelf ℝ E) φ y)) u) v

Local solutions of the conformal flattening equation. Let g be a smooth Riemannian metric on a finite-dimensional real normed space E whose Weyl and Cotton tensors vanish near x₀. Then for every c : ℝ and every tangent vector ξ at x₀ there is a smooth function φ with φ x₀ = c and grad φ x₀ = ξ solving the conformal flattening equation Hess φ = P + dφ ⊗ dφ - (|dφ|² / 2) g near x₀, where P is the Schouten tensor. When E has dimension at least 3, this says equivalently that e^(2φ) g has vanishing Schouten tensor near x₀.