The Poincaré–Wirtinger inequality on a closed Riemannian manifold

Let M be a compact connected manifold without boundary with a continuous Riemannian metric and its Riemannian volume dV. For 1 ≤ p < ∞, the Poincaré–Wirtinger inequality says that there is a constant C with

‖u - ⨍ u‖_{Lᵖ} ≤ C ‖∇u‖_{Lᵖ}

for every C¹ function u, where ⨍ u is the mean of u with respect to dV (TauCeti.Manifold.exists_eLpNorm_sub_average_le_mul_eLpNorm_mgradient). It is the coercivity estimate of variational problems on M normalized by the mean: on functions of mean zero the Dirichlet energy ∫ |∇u|² dV controls the full H¹ norm. This is the estimate needed to invert the Laplacian on functions of mean zero, and to minimize functionals such as ∫ (|∇u|² / 2 + K u + e^(2u) / 2) dV, whose critical points solve the Gaussian curvature equation Δ⁺u + K + e^(2u) = 0 on a closed surface.

The proof first localizes the Poincaré–Wirtinger inequality on a ball of the model space, TauCeti.lintegral_enorm_sub_setAverage_rpow_le_of_convex. Near each point c, in a coordinate ball around the chart image of c, the chart density is comparable to its value at c and the coordinate derivative of every function is bounded by a fixed multiple of the length of its gradient, so on the chart preimage of the ball u is within a fixed multiple of ‖∇u‖_{Lᵖ(M)} of a constant in Lᵖ (TauCeti.Manifold.exists_isOpen_eLpNorm_restrict_sub_le_mul_eLpNorm_mgradient). Finitely many such neighbourhoods cover M, and since M is connected the local constants chain together to a single one (TauCeti.exists_eLpNorm_sub_const_le_of_isOpen_cover). Replacing that constant by the mean costs at most a factor 2 (TauCeti.eLpNorm_sub_setAverage_le).

Connectedness cannot be dropped: on a disjoint union of two closed manifolds a function equal to different constants on the two components has vanishing gradient but is not equal to its mean.

Main results

• TauCeti.Manifold.exists_isOpen_eLpNorm_restrict_sub_le_mul_eLpNorm_mgradient: the local local estimate, near any point of a manifold without boundary, of the Lᵖ oscillation of u on a neighbourhood V by the Lᵖ norm of ∇u over all of M.
• TauCeti.Manifold.exists_eLpNorm_sub_average_le_mul_eLpNorm_mgradient: the Poincaré–Wirtinger inequality on a closed connected Riemannian manifold.
• TauCeti.Manifold.exists_integral_sq_sub_average_le_mul_integral_norm_sq_mgradient: its L² case in integral form, ∫ (u - ⨍ u)² dV ≤ C ∫ ‖∇u‖² dV.

References

• T. Aubin, Some Nonlinear Problems in Riemannian Geometry, Springer (1998), Chapter 2.
• E. Hebey, Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities, Courant Lecture Notes 5 (1999), Chapter 3.
• D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer (2001), Lemma 7.16 and §7.8.

theorem TauCeti.Manifold.exists_isOpen_eLpNorm_restrict_sub_le_mul_eLpNorm_mgradient {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] [ChartedSpace H M] [IsManifold I 1 M] [Bundle.RiemannianBundle fun (x : M) => TangentSpace I x] [IsContinuousRiemannianBundle E fun (x : M) => TangentSpace I x] [MeasurableSpace M] [BorelSpace M] [LindelofSpace M] {p : ENNReal} (hp : 1 ≤ p) (hp' : p ≠ ⊤) (c : M) : ∃ (V : Set M), IsOpen V ∧ c ∈ V ∧ ∃ (C : NNReal), ∀ (u : M → ℝ), ContMDiff I (modelWithCornersSelf ℝ ℝ) 1 u → ∃ (a : ℝ), MeasureTheory.eLpNorm (fun (x : M) => u x - a) p ((riemannianVolume I M).restrict V) ≤ ↑C * MeasureTheory.eLpNorm (fun (x : M) => ‖mgradient I u x‖) p (riemannianVolume I M)

A local oscillation bound near a point. Let 1 ≤ p < ∞, on a manifold without boundary with a continuous Riemannian metric. Every point c lies in an open set V with a constant C such that every C¹ function u satisfies ‖u - a‖_{Lᵖ(V)} ≤ C ‖∇u‖_{Lᵖ(M)} for some constant a. The gradient norm on the right is taken over all of M, not only over V.

theorem TauCeti.Manifold.exists_eLpNorm_sub_average_le_mul_eLpNorm_mgradient {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] [ChartedSpace H M] [IsManifold I 1 M] [Bundle.RiemannianBundle fun (x : M) => TangentSpace I x] [IsContinuousRiemannianBundle E fun (x : M) => TangentSpace I x] [MeasurableSpace M] [BorelSpace M] [CompactSpace M] [PreconnectedSpace M] {p : ENNReal} (hp : 1 ≤ p) (hp' : p ≠ ⊤) : ∃ (C : NNReal), ∀ (u : M → ℝ), ContMDiff I (modelWithCornersSelf ℝ ℝ) 1 u → MeasureTheory.eLpNorm (fun (x : M) => u x - ⨍ (y : M), u y ∂riemannianVolume I M) p (riemannianVolume I M) ≤ ↑C * MeasureTheory.eLpNorm (fun (x : M) => ‖mgradient I u x‖) p (riemannianVolume I M)

The Poincaré–Wirtinger inequality on a closed Riemannian manifold. Let 1 ≤ p < ∞, on a compact connected manifold without boundary with a continuous Riemannian metric. There is a constant C such that ‖u - ⨍ u‖_{Lᵖ} ≤ C ‖∇u‖_{Lᵖ} for every C¹ function u, where ⨍ u is the mean of u with respect to the Riemannian volume.

theorem TauCeti.Manifold.exists_integral_sq_sub_average_le_mul_integral_norm_sq_mgradient {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] [ChartedSpace H M] [IsManifold I 1 M] [Bundle.RiemannianBundle fun (x : M) => TangentSpace I x] [IsContinuousRiemannianBundle E fun (x : M) => TangentSpace I x] [MeasurableSpace M] [BorelSpace M] [CompactSpace M] [PreconnectedSpace M] : ∃ (C : NNReal), ∀ (u : M → ℝ), ContMDiff I (modelWithCornersSelf ℝ ℝ) 1 u → ∫ (x : M), (u x - ⨍ (y : M), u y ∂riemannianVolume I M) ^ 2 ∂riemannianVolume I M ≤ ↑C * ∫ (x : M), ‖mgradient I u x‖ ^ 2 ∂riemannianVolume I M

The L² Poincaré–Wirtinger inequality in integral form. On a compact connected manifold without boundary with a continuous Riemannian metric, there is a constant C such that ∫ (u - ⨍ u)² dV ≤ C ∫ ‖∇u‖² dV for every C¹ function u.