Hessians under conformal changes of metric

For a positive factor f, the scalar Hessian for f g differs from that for g by -(2 f)⁻¹ (df ⊗ du + du ⊗ df - ⟪grad f, grad u⟫ g). This is a tensor identity, not just its trace: it controls second directional derivatives in a conformally changed metric. We also give the real-power version, which applies to the factor u ^ (4 / (n - 2)). Taking the trace gives the Laplacian of f g in real dimension n: Δ_{f g} u = f⁻¹ (Δ_g u + (n - 2) / (2 f) ⟪grad_g f, grad_g u⟫_g). The metric needs only one continuous derivative, and the differentiated function needs two continuous derivatives at the point in question.

The two metric instances in each statement have disjoint scopes. All derivatives and tangent vectors use the native manifold carriers. The calculation combines TauCeti.Manifold.hessian_apply_eq_mvfderiv_sub with Bundle.ContMDiffRiemannianMetric.leviCivitaConnection_rescale_apply.

References

• J. M. Lee and T. H. Parker, The Yamabe problem, Bull. Amer. Math. Soc. 17 (1987), §2.
• A. L. Besse, Einstein Manifolds, §1.J, for the conformal connection formula.

theorem Bundle.ContMDiffRiemannianMetric.hessian_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] [IsManifold I 2 M] {n : WithTop ℕ∞} (g : ContMDiffRiemannianMetric I n E fun (x : M) => TangentSpace I x) (hn : 1 ≤ n) {f : M → ℝ} (hf : ContMDiff I (modelWithCornersSelf ℝ ℝ) n f) (hf_pos : ∀ (x : M), 0 < f x) {u : M → ℝ} {x : M} (hu : ContMDiffAt I (modelWithCornersSelf ℝ ℝ) 2 u x) (v w : TangentSpace I x) : TauCeti.Manifold.hessian I u x v w = TauCeti.Manifold.hessian I u x v w - (2 * f x)⁻¹ * ((d% f x) v * (d% u x) w + (d% f x) w * (d% u x) v - Inner.inner ℝ (TauCeti.Manifold.mgradient I f x) (TauCeti.Manifold.mgradient I u x) * Inner.inner ℝ v w)

Conformal Hessian formula. For ĝ = f g, Hess_ĝ u = Hess_g u - (2 f)⁻¹ (df ⊗ du + du ⊗ df - ⟪grad f, grad u⟫_g g).

theorem Bundle.ContMDiffRiemannianMetric.hessian_rescale_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] [IsManifold I 2 M] {n : WithTop ℕ∞} (g : ContMDiffRiemannianMetric I n E fun (x : M) => TangentSpace I x) (hn : 1 ≤ n) {a : M → ℝ} (ha : ContMDiff I (modelWithCornersSelf ℝ ℝ) n a) (ha_pos : ∀ (x : M), 0 < a x) (p : ℝ) {hf : ContMDiff I (modelWithCornersSelf ℝ ℝ) n fun (x : M) => a x ^ p} {hf_pos : ∀ (x : M), 0 < a x ^ p} {u : M → ℝ} {x : M} (hu : ContMDiffAt I (modelWithCornersSelf ℝ ℝ) 2 u x) (v w : TangentSpace I x) : TauCeti.Manifold.hessian I u x v w = TauCeti.Manifold.hessian I u x v w - p / (2 * a x) * ((d% a x) v * (d% u x) w + (d% a x) w * (d% u x) v - Inner.inner ℝ (TauCeti.Manifold.mgradient I a x) (TauCeti.Manifold.mgradient I u x) * Inner.inner ℝ v w)

Hessian under a power rescaling. For ĝ = a ^ p g with a > 0, the Hessian correction has coefficient p / (2 a). In particular p = 4 / (dim M - 2) gives the Yamabe rescaling coefficient 2 / ((dim M - 2) a).

theorem Bundle.ContMDiffRiemannianMetric.laplaceBeltrami_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] [IsManifold I 2 M] {n : WithTop ℕ∞} (g : ContMDiffRiemannianMetric I n E fun (x : M) => TangentSpace I x) (hn : 1 ≤ n) {f : M → ℝ} (hf : ContMDiff I (modelWithCornersSelf ℝ ℝ) n f) (hf_pos : ∀ (x : M), 0 < f x) {u : M → ℝ} {x : M} (hu : ContMDiffAt I (modelWithCornersSelf ℝ ℝ) 2 u x) : TauCeti.Manifold.laplaceBeltrami I u x = (f x)⁻¹ * (TauCeti.Manifold.laplaceBeltrami I u x + (↑(Module.finrank ℝ E) - 2) / (2 * f x) * Inner.inner ℝ (TauCeti.Manifold.mgradient I f x) (TauCeti.Manifold.mgradient I u x))

Laplacian under conformal change. For ĝ = f g on a manifold of dimension n, Δ_ĝ u = f⁻¹ (Δ_g u + (n - 2) / (2 f) ⟪grad_g f, grad_g u⟫_g), with Δ = div grad. Only C¹ regularity of the metric and factor and pointwise C² regularity of the function are required.