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Divergence comparison in a smooth compact boundary collar

Statement

Work in ZFC. Let $(N, h)$ be a smooth Riemannian three-manifold with compact boundary $S$ , and suppose a smooth collar diffeomorphism $F : S \times [0, η_{0}] \to U \subset N$ satisfies $F (p, 0) = p$ and $ρ (F (p, t)) = t$ , where $∣ \nabla ρ ∣_{h} = 1$ and $∣ Δ_{h} ρ ∣ \leq C$ on $U$ , for constants $η_{0} > 0$ and $C \geq 0$ . Here $\nabla ρ$ is the metric gradient and $Δ_{h} ρ = {div}_{h} (\nabla ρ)$ ; area and volume are integrals of the induced Riemannian densities.

For $0 < η \leq η_{0}$ , let $Ω \subset U \cap {0 \leq ρ \leq η}$ be a compact smooth three-dimensional region with boundary the disjoint union $S ⊔ D$ , and assume that $Ω$ contains the inward side of $S$ . Here a smooth region is supplied as an embedded smooth submanifold with boundary of $N$ , whose inclusion $ι : Ω ↪ N$ is a smooth embedding and whose metric is $ι^{*} h$ . Its intrinsic interior $Ω^{\circ}$ is open in $N^{\circ}$ with the restricted smooth structure, and its displayed boundary is its intrinsic manifold boundary. Containing the inward side means that a neighborhood of $S$ in $N$ lies in $Ω$ , with the restricted manifold-with-boundary smooth structure there. The outer boundary $D$ may be disconnected. Then $∣ S ∣ \leq ∣ D ∣ + C {Vol}_{h} (Ω)$ .

If the collar volume density has the form $F^{*} d V_{h} = J (p, t) d A_{S} (p) d t$ with $J \leq K$ for a constant $K$ , then ${Vol}_{h} (Ω) \leq K η ∣ S ∣$ , so $∣ S ∣ \leq ∣ D ∣ + C K η ∣ S ∣$ . In particular, for fixed collar data the error is $O (η)$ , uniformly over such regions.

Facts & Assumptions

[A1]

ZFC supplies the countable choice required by the cited integration, measure and partition results. The Axiom of Choice

[F1]

A Riemannian metric is smooth and positive definite; the gradient is the metric dual of the differential, and the Laplace–Beltrami operator is the trace of the Hessian. Its value is the divergence of the gradient. Riemannian divergence Riemannian metric and riemannian manifold The Riemannian gradient is the metric dual of the differential Laplace–Beltrami operator as the trace of the Hessian Gradient hessian and divergence connection formulas

[F2]

Riemannian densities are defined locally by the square root of the metric determinant. Compactly supported density integration is local, linear, nonnegative and invariant under diffeomorphisms, without an orientation. Riemannian volume density Orientation-free density integration and its properties

[F3]

On an oriented Riemannian manifold with boundary, the integral of the divergence of a smooth compactly supported vector field equals its outward normal flux. A smooth partition of unity subordinate to a boundary-chart cover exists under countable choice. Riemannian divergence theorem Smooth partitions of unity exist on manifolds with boundary

[F4]

Cauchy–Schwarz bounds the absolute inner product by the product of the norms. Tonelli's theorem permits iterated integration of a nonnegative measurable function over a product of sigma-finite measure spaces. Cauchy–Schwarz: |⟨ u,v⟩|≤‖ u‖‖ v‖, with equality exactly for linearly dependent vectors Tonelli and Fubini for the completed product, with only almost-everywhere section measurability

[F5]

The boundary of a smooth positive-dimensional manifold is a closed embedded smooth boundaryless hypersurface, and its tangent space is the boundary-tangent hyperplane. Pullback of a Riemannian metric by an immersion is Riemannian. The boundary of a positive-dimensional manifold is a closed embedded smooth (n-1)-manifold The boundary tangent space is the boundary-tangent hyperplane Pullback of a riemannian metric is riemannian exactly for immersions

[F6]

Under countable choice a positive smooth density defines a compact-finite, sigma-finite Borel measure. On a Borel subset of a chart its measure is the Lebesgue integral of the density coefficient. For a nonnegative Borel function the integral is the sum of the partition-weighted chart integrals; for a smooth compactly supported function it agrees with smooth density integration. Positive smooth densities give Radon volume Intrinsic density measure and its chart restriction Measurable integration extends smooth density integration

[F7]

Euclidean Lebesgue measures are sigma-finite, and the product of the measures in dimensions two and one agrees with three-dimensional Lebesgue measure on Borel sets. Lebesgue measure is sigma-finite, and every metrically bounded subset of ℝⁿ has finite outer measure On Borel subsets of ℝᵐ⁺ⁿ, the product λₘ × λₙ agrees with λₘ₊ₙ

[F8]

Under countable choice a positive-codimension immersed submanifold of a smooth manifold is null in ambient coordinates. Thus its Borel image has zero measure for a smooth density by the chart formula in [F6]. Positive-codimension immersed submanifolds are null

[F9]

Nonnegative measurable integrals are monotone and homogeneous for positive finite scalars; the zero function has integral zero. Monotonicity and nonnegative homogeneity of the nonnegative integral

[F10]

A diffeomorphism is a bijective smooth map with smooth inverse. This definition applies to the supplied closed-interval parametrization onto its smooth image $U$ . Diffeomorphisms and local diffeomorphisms of manifolds

[F11]

Under countable choice, every closed interval $[a, b]$ with real $a \leq b$ has one-dimensional Lebesgue measure $b - a$ . In particular, the degenerate interval $[c, c] = {c}$ is Borel and has measure zero for every real $c$ . A box in ℝⁿ with parameters aᵢ≤ bᵢ is Lebesgue measurable of measure ∏_i<n(bᵢ-aᵢ), whichever of its faces are included

[F12]

An embedded smooth submanifold with boundary is a subset supplied with a manifold-with-boundary smooth structure for which its inclusion is a smooth embedding. An embedding is an immersion and a homeomorphism onto its image; its differential is injective on the full tangent spaces, including at boundary points. This definition does not assert that the intrinsic boundary equals the intersection with the ambient boundary. Embedded smooth submanifolds with boundary Immersions and embeddings for manifolds with boundary

Proof

Given: The collar, region and bounds in the statement, working in ZFC as in [A1].

Goal: Prove the area inequality and the stated uniform volume estimate.

1.1A1F1F2F5F12given

If $S$ is empty, the collar and $Ω$ are empty and all conclusions hold with zero integrals. Otherwise [F12] makes the inclusion $ι : Ω ↪ N$ a smooth embedding and hence an immersion. Apply [F5] to $N$ and to the supplied manifold with boundary $Ω$ . It gives the smooth embedded surfaces $S$ and $D$ and identifies their tangent spaces with the boundary hyperplanes of $Ω$ ; at $S$ the stated region convention gives the same smooth structure as the boundary of $N$ . Their inclusions into $N$ are immersions, so the induced metrics and their area densities are smooth and positive. The metric on $Ω$ is $ι^{*} h$ , smooth and positive up to its boundary by [F5].

2.1A1F1F2F3step 1.1given

Cover $Ω$ by coordinate neighborhoods in the manifold with boundary $Ω$ and take a subordinate smooth partition of unity. Compactness and local finiteness leave finitely many nonzero terms $χ_{i}$ , each supported compactly inside its chart. Orient each chart and apply the oriented divergence theorem to $χ_{i} \nabla ρ$ there. On that oriented chart the volume and outward boundary forms integrate as the corresponding positive densities. Sum using locality, linearity and orientation independence of density integration. The product rule and $\sum_{i} χ_{i} = 1$ give $\sum_{i} div (χ_{i} \nabla ρ) = div (\nabla ρ) = Δ_{h} ρ$ . Hence $\int_{Ω} Δ_{h} ρ d V_{h} = \int_{\partial Ω} h (\nabla ρ, ν) d A$ , where $ν$ is the outward unit normal. This also covers nonorientable $Ω$ .

2.2A1F4F5F6F7F8F10F11F12step 1.1given

Write $B_{η} = F (S \times [0, η])$ . Let $μ$ be the Borel measure of the smooth positive density $d V_{h}$ on $N$ supplied by [F6]; $B_{η}$ and $Ω$ are compact and hence Borel, and $μ (B_{η}) < \infty$ . The intrinsic smooth volume of each region equals its ambient density measure. Indeed, for $Ω$ the stated region convention identifies the interior smooth structure with the ambient restriction, and its metric is the pullback through its embedding as in step 1.1; for $B_{η}$ use the supplied smooth collar image and [F10]. Thus on each interior the charts and density coefficients agree, so [F6] gives the same measure there. The remaining boundary has zero three-dimensional measure intrinsically: in a boundary chart it is a Borel coordinate face, each of whose normal sections is either empty or a singleton. By [F11], $λ_{1} ({c}) = λ_{1} ([c, c]) = 0$ for every real $c$ . The sigma-finite factors and Euclidean product identification in [F7] therefore allow Tonelli from [F4] to give zero three-dimensional measure for the face. The same is true in ambient boundary charts for $S$ . The surface $D$ lies in the interior of $N$ and is an immersed two-manifold there, so [F8] makes it ambient-null; the same argument applies to the upper collar surface $F (S \times {η})$ . Coordinate-null Borel sets have zero smooth-density measure by [F6]. Finally apply the smooth-integral agreement of [F6] to the constant function one on each compact region. Since $Ω \subset B_{η}$ , measure additivity and nonnegativity give ${Vol}_{h} (Ω) = μ (Ω) \leq μ (B_{η}) = {Vol}_{h} (B_{η})$ .

3.1F1F2F4F5step 1.1step 2.1given

At $S$ the restriction of $ρ$ is zero, so its differential vanishes on the tangent hyperplane from step 1.1. The gradient-dual identity makes $\nabla ρ$ perpendicular to that hyperplane. It is unit and points inward: $ρ$ is nonnegative on $Ω$ , and $d ρ (\nabla ρ) = 1$ , with $Ω$ containing the inward side of $S$ . Thus $h (\nabla ρ, ν) = - 1$ at $S$ . On $D$ , Cauchy–Schwarz and $∣ \nabla ρ ∣ = ∣ ν ∣ = 1$ imply $h (\nabla ρ, ν) \leq 1$ . The identity in step 2.1 and $∣ Δ_{h} ρ ∣ \leq C$ now yield $∣ S ∣ = \int_{D} h (\nabla ρ, ν) d A - \int_{Ω} Δ_{h} ρ d V_{h} \leq ∣ D ∣ + C {Vol}_{h} (Ω)$ .

3.2A1F2F3F4F6F7F10step 2.2given

To compute this last volume assume the stated factorization $F^{*} d V_{h} = J d A_{S} d t$ . Its coefficient $J$ is smooth and positive, being the ratio of positive smooth densities. Choose a finite smooth chart partition $(x_{i}, ψ_{i})$ on the compact surface $S$ , with $x_{i} : V_{i} \to x_{i} (V_{i}) \subset R^{2}$ . Write $d A_{S} = a_{i} (u) ∣ d u^{1} d u^{2} ∣$ in this chart. On $S \times (0, η)$ use the product charts and the partition $ψ_{i} (p)$ , and extend each coefficient $g_{i} (u, t) = ψ_{i} (x_{i}^{- 1} u) a_{i} (u) J (x_{i}^{- 1} u, t)$ by zero outside $x_{i} (V_{i}) \times (0, η)$ . Each extension is nonnegative Borel: its coefficient is smooth on an open set, and the extension is zero on its Borel complement. The measures $λ_{2}$ and $λ_{1}$ are sigma-finite by [F7], so Tonelli applies to these Borel functions, also on the completed product. The product identification in [F7] equates their three-dimensional chart integrals with the product-measure integrals. The end surfaces have zero volume by step 2.2. Density diffeomorphism invariance, the chart formula of [F6], and the finite partition sum consequently give ${Vol}_{h} (B_{η}) = \sum_{i} \int_{0}^{η} \int_{x_{i} (V_{i})} g_{i} (u, t) d λ_{2} (u) d t = \int_{0}^{η} \int_{S} J (p, t) d μ_{S} (p) d t$ , where $μ_{S}$ is the finite Borel measure of $d A_{S}$ and $μ_{S} (S) = ∣ S ∣$ . This derives the iterated-integral formula from chart integrals rather than assuming a product-measure identity for manifold densities.

4.1F6F9step 3.1step 2.2step 3.2given∎

By $J \leq K$ and [F9], the inner integral in step 3.2 is at most $K μ_{S} (S) = K ∣ S ∣$ . All quantities are finite by compactness and [F6]; thus integrating over an interval of length $η$ gives ${Vol}_{h} (B_{η}) \leq K η ∣ S ∣$ . Combining steps 2.2 and 3.1 proves $∣ S ∣ \leq ∣ D ∣ + C K η ∣ S ∣$ . With the collar, $C, K$ and $∣ S ∣$ fixed, this error is $O (η)$ independently of the smooth region or its disconnected outer boundary.

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