Total curvature of a closed surface under conformal change

Let g be a smooth Riemannian metric on a closed surface M and ĝ = f g a conformal metric, with f = e^(2φ). In dimension two the volume of ĝ has density f against that of g, and the conformal change of scalar curvature reads f Scal^ĝ = Scal^g - 2 Δφ, with Δ = div grad the Laplace–Beltrami operator of g. Since ∫ Δφ dV_g = 0 on a closed manifold, the total curvature is a conformal invariant:

∫ Scal^ĝ dV_ĝ = ∫ Scal^g dV_g.

When ĝ has constant scalar curvature c, the left side is c Vol(ĝ), which gives the area normalization c Vol(ĝ) = ∫ Scal^g dV_g. For a metric of Gaussian curvature -1, that is c = -2, this says that its area is -∫ K dV_g with K = Scal^g / 2, and that a closed surface whose conformal class contains a metric of constant negative curvature has negative total curvature.

As in TauCeti.WithMetric.integral_riemannianVolume_rescale, the volumes of g and ĝ are those of the metric-indexed copies TauCeti.WithMetric g and TauCeti.WithMetric ĝ of M.

Main results

• Bundle.ContMDiffRiemannianMetric.integral_scalarCurvature_rescale_of_finrank_eq_two: the total curvature of a closed surface is invariant under conformal change.
• Bundle.ContMDiffRiemannianMetric.mul_riemannianTotalVolume_rescale_of_finrank_eq_two: a conformal metric of constant scalar curvature c has area Vol(ĝ) with c Vol(ĝ) = ∫ Scal^g dV_g.
• Bundle.ContMDiffRiemannianMetric.integral_scalarCurvature_neg_of_scalarCurvature_rescale_eq: a nonempty closed surface conformal to a metric of constant negative curvature has negative total curvature.

References

• J. L. Kazdan and F. W. Warner, Curvature functions for compact 2-manifolds, Ann. of Math. 99 (1974), 14–47.
• M. S. Berger, Riemannian structures of prescribed Gaussian curvature for compact 2-manifolds, J. Differential Geom. 5 (1971), 325–332.

theorem Bundle.ContMDiffRiemannianMetric.integral_scalarCurvature_rescale_of_finrank_eq_two {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] [T2Space M] [CompactSpace M] [ChartedSpace H M] [IsManifold I (↑⊤) M] (g : ContMDiffRiemannianMetric I (↑⊤) E fun (x : M) => TangentSpace I x) (hE : Module.finrank ℝ E = 2) {f : M → ℝ} (hf : ContMDiff I (modelWithCornersSelf ℝ ℝ) (↑⊤) f) (hf_pos : ∀ (x : M), 0 < f x) : ∫ (x : TauCeti.WithMetric (g.rescale f hf hf_pos)), (g.rescale f hf hf_pos).scalarCurvature x.ofWithMetric ∂TauCeti.riemannianVolume I (TauCeti.WithMetric (g.rescale f hf hf_pos)) = ∫ (x : TauCeti.WithMetric g), g.scalarCurvature x.ofWithMetric ∂TauCeti.riemannianVolume I (TauCeti.WithMetric g)

Total curvature is a conformal invariant of a closed surface. For a smooth positive function f on a closed surface, ∫ Scal^(f g) dV_(f g) = ∫ Scal^g dV_g.

theorem Bundle.ContMDiffRiemannianMetric.mul_riemannianTotalVolume_rescale_of_finrank_eq_two {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] [T2Space M] [CompactSpace M] [ChartedSpace H M] [IsManifold I (↑⊤) M] (g : ContMDiffRiemannianMetric I (↑⊤) E fun (x : M) => TangentSpace I x) (hE : Module.finrank ℝ E = 2) {f : M → ℝ} (hf : ContMDiff I (modelWithCornersSelf ℝ ℝ) (↑⊤) f) (hf_pos : ∀ (x : M), 0 < f x) {c : ℝ} (h : ∀ (x : M), (g.rescale f hf hf_pos).scalarCurvature x = c) : c * TauCeti.riemannianTotalVolume I (TauCeti.WithMetric (g.rescale f hf hf_pos)) = ∫ (x : TauCeti.WithMetric g), g.scalarCurvature x.ofWithMetric ∂TauCeti.riemannianVolume I (TauCeti.WithMetric g)

Area normalization. On a closed surface, a conformal metric f g of constant scalar curvature c satisfies c Vol(f g) = ∫ Scal^g dV_g. For a metric of Gaussian curvature -1, c = -2, its area is -∫ K dV_g with K = Scal^g / 2.

theorem Bundle.ContMDiffRiemannianMetric.integral_scalarCurvature_neg_of_scalarCurvature_rescale_eq {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] [T2Space M] [CompactSpace M] [ChartedSpace H M] [IsManifold I (↑⊤) M] (g : ContMDiffRiemannianMetric I (↑⊤) E fun (x : M) => TangentSpace I x) [Nonempty M] (hE : Module.finrank ℝ E = 2) {f : M → ℝ} (hf : ContMDiff I (modelWithCornersSelf ℝ ℝ) (↑⊤) f) (hf_pos : ∀ (x : M), 0 < f x) {c : ℝ} (hc : c < 0) (h : ∀ (x : M), (g.rescale f hf hf_pos).scalarCurvature x = c) : ∫ (x : TauCeti.WithMetric g), g.scalarCurvature x.ofWithMetric ∂TauCeti.riemannianVolume I (TauCeti.WithMetric g) < 0

Sign of the total curvature. If a metric g on a nonempty closed surface is conformal to a metric of constant negative scalar curvature, then the total curvature of g is negative.