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Formal adjoint of a real drift Laplacian on a closed manifold

Statement

Let $(S, h)$ be a connected compact smooth Riemannian manifold without boundary, of positive dimension. For a smooth real vector field $t$ and smooth real function $c$ , put $L u = - Δ_{h} u + 2 ⟨ t, \nabla u ⟩_{h} + c u$ , with $Δ_{h} = {div}_{h} {grad}_{h}$ . The formal adjoint for $⟨ f, g ⟩_{L^{2}} = \int_{S} \overline{f} g d μ_{h}$ is $L^{†} u = - Δ_{h} u - 2 ⟨ t, \nabla u ⟩_{h} + (c - 2 {div}_{h} t) u$ . It has smooth real coefficients and the same negative-Laplacian principal part. The measure $d μ_{h}$ is the positive Riemannian density, so no orientation assumption is needed.

Facts & Assumptions

Given: the hypotheses in the Statement; work in ZFC.

Goal: establish the Statement.

[F1]

The supplied metric has its Levi–Civita connection. Gradient and Laplacian use ${grad}_{h} u = (d u)^{♯}$ and $Δ_{h} = {div}_{h} {grad}_{h}$ ; in coordinates $({grad}_{h} u)^{i} = h^{i j} \partial_{j} u$ and ${div}_{h} X = γ^{- 1} \partial_{i} (γ X^{i})$ , where $γ = \sqrt{\det (h_{i j})}$ . Differential expressions extend complex-linearly, and contractions in them use the complex-bilinear extension of $h$ . Riemannian metric and riemannian manifold Fundamental theorem of riemannian geometry Riemannian gradient Laplace–Beltrami operator as the trace of the Hessian Coordinate formula for riemannian divergence

[F2]

Density frames transform by absolute Jacobians. The local expressions $d μ_{h} = \sqrt{\det (h_{i j})} ∣ d x^{1} \dots d x^{n} ∣$ therefore glue to a smooth positive density on $S$ , independently of orientability. Density bundle and smooth density fields Riemannian volume density The riemannian volume density is coordinate independent

[F3]

Compactly supported smooth density integration is defined by chart partitions, is independent of those choices, is linear and local, and is strictly positive on any nonzero nonnegative density. Integrate complex coefficients by their real and imaginary parts. Compactness makes all smooth function multiples of $d μ_{h}$ compactly supported. Work in ZFC: Choice supplies the countable-choice hypotheses of density integration and partitions. The $L^{2}$ notation here denotes the displayed pairing on smooth functions. Integral of a compactly supported smooth density Orientation-free density integration and its properties The Axiom of Choice

[F4]

A smooth partition of unity subordinate to a coordinate cover exists under countable choice. The divergence theorem is applied only on an oriented coordinate domain, with its restricted metric and a compactly supported vector field; that domain has no manifold boundary, so its divergence integral is zero. No orientation of $S$ is assumed. Smooth partitions of unity exist on manifolds with boundary Riemannian divergence theorem The Axiom of Choice

[F5]

Here a formal adjoint of $L$ means a linear smooth-coefficient differential operator $A$ satisfying $⟨ f, L g ⟩_{L^{2}} = ⟨ A f, g ⟩_{L^{2}}$ for every $f, g \in C^{\infty} (S; C)$ . This is an identity of the density integrals just specified. Existence and uniqueness for this operator are proved below. Integral of a compactly supported smooth density Orientation-free density integration and its properties

[F6]

For scalar coordinate operators built from the supplied gradient and divergence, use this symbol convention: for second-order part $- a^{i j} \partial_{i} \partial_{j}$ , put $σ_{2} (P) (x, ξ) = a^{i j} (x) ξ_{i} ξ_{j}$ . Thus the sign is the one obtained by replacing each top-order derivative by $i ξ_{j}$ . This convention applies in every positive dimension; its value for the two operators is calculated in the proof. Riemannian metric and riemannian manifold Riemannian gradient Coordinate formula for riemannian divergence

Proof

Given: the hypotheses in the Statement; work in ZFC.

Goal: establish the Statement.

[F1] The supplied metric has its Levi–Civita connection. Gradient and Laplacian use ${grad}_{h} u = (d u)^{♯}$ and $Δ_{h} = {div}_{h} {grad}_{h}$ ; in coordinates $({grad}_{h} u)^{i} = h^{i j} \partial_{j} u$ and ${div}_{h} X = γ^{- 1} \partial_{i} (γ X^{i})$ , where $γ = \sqrt{\det (h_{i j})}$ . Differential expressions extend complex-linearly, and contractions in them use the complex-bilinear extension of $h$ . Riemannian metric and riemannian manifold Fundamental theorem of riemannian geometry Riemannian gradient Laplace–Beltrami operator as the trace of the Hessian Coordinate formula for riemannian divergence

[F2] Density frames transform by absolute Jacobians. The local expressions $d μ_{h} = \sqrt{\det (h_{i j})} ∣ d x^{1} \dots d x^{n} ∣$ therefore glue to a smooth positive density on $S$ , independently of orientability. Density bundle and smooth density fields Riemannian volume density The riemannian volume density is coordinate independent

[F3] Compactly supported smooth density integration is defined by chart partitions, is independent of those choices, is linear and local, and is strictly positive on any nonzero nonnegative density. Integrate complex coefficients by their real and imaginary parts. Compactness makes all smooth function multiples of $d μ_{h}$ compactly supported. Work in ZFC: Choice supplies the countable-choice hypotheses of density integration and partitions. The $L^{2}$ notation here denotes the displayed pairing on smooth functions. Integral of a compactly supported smooth density Orientation-free density integration and its properties The Axiom of Choice

[F4] A smooth partition of unity subordinate to a coordinate cover exists under countable choice. The divergence theorem is applied only on an oriented coordinate domain, with its restricted metric and a compactly supported vector field; that domain has no manifold boundary, so its divergence integral is zero. No orientation of $S$ is assumed. Smooth partitions of unity exist on manifolds with boundary Riemannian divergence theorem The Axiom of Choice

[F5] Here a formal adjoint of $L$ means a linear smooth-coefficient differential operator $A$ satisfying $⟨ f, L g ⟩_{L^{2}} = ⟨ A f, g ⟩_{L^{2}}$ for every $f, g \in C^{\infty} (S; C)$ . This is an identity of the density integrals just specified. Existence and uniqueness for this operator are proved below. Integral of a compactly supported smooth density Orientation-free density integration and its properties

[F6] For scalar coordinate operators built from the supplied gradient and divergence, use this symbol convention: for second-order part $- a^{i j} \partial_{i} \partial_{j}$ , put $σ_{2} (P) (x, ξ) = a^{i j} (x) ξ_{i} ξ_{j}$ . Thus the sign is the one obtained by replacing each top-order derivative by $i ξ_{j}$ . This convention applies in every positive dimension; its value for the two operators is calculated in the proof. Riemannian metric and riemannian manifold Riemannian gradient Coordinate formula for riemannian divergence

Proof technique: direct.

1.1F1F2F3F4

Choose a smooth chart-subordinate partition $(ρ_{a})$ . Local finiteness and compactness make only finitely many supports nonempty: finitely many neighborhoods covering $S$ each meet only finitely many supports. Each $ρ_{a} X$ is compactly supported inside its chart domain $U_{a}$ , which is oriented by its coordinates. There $d μ_{h}$ and the positive coordinate Riemannian volume form have the same coefficient $γ$ , so their integrals of a chart-supported scalar agree. The oriented divergence theorem on $U_{a}$ gives $\int_{S} {div}_{h} (ρ_{a} X) d μ_{h} = 0$ , using locality to extend the chart integral to $S$ . The coordinate product rule gives ${div}_{h} (ρ_{a} X) = ρ_{a} {div}_{h} X + d ρ_{a} (X)$ . Sum the finite family, using $\sum_{a} ρ_{a} = 1$ and $\sum_{a} d ρ_{a} = 0$ , to obtain $\int_{S} {div}_{h} X d μ_{h} = 0$ . The same identity holds for complex vector fields by real and imaginary parts.

2.1F1F3F5step 1.1

Apply step 1.1 to $\overline{f} {grad}_{h} g$ and then to $g {grad}_{h} \overline{f}$ . The product rule gives $\int_{S} \overline{f} (- Δ_{h} g) d μ_{h} = \int_{S} h (\nabla \overline{f}, \nabla g) d μ_{h} = \int_{S} \overline{(- Δ_{h} f)} g d μ_{h}$ . Thus the negative Laplacian is formally symmetric for the stated first-variable-conjugate-linear pairing.

3.1F1F3F5step 1.1step 2.1

Apply step 1.1 to $\overline{f} g t$ . Expanding its divergence yields $\int_{S} \overline{f} 2 t \cdot \nabla g d μ_{h} = - \int_{S} g (2 t \cdot \nabla \overline{f} + 2 {div}_{h} t \overline{f}) d μ_{h}$ . Combining this with step 2.1 and the real multiplication coefficient $c$ proves $⟨ f, L g ⟩_{L^{2}} = ⟨ A f, g ⟩_{L^{2}}$ for $A f = - Δ_{h} f - 2 t \cdot \nabla f + (c - 2 {div}_{h} t) f$ . All these coefficients are smooth and real.

4.1F2F3F5step 3.1

If $A$ and $B$ both satisfy the identity in F5, then for each smooth $f$ and every smooth $g$ one has $⟨ (A - B) f, g ⟩_{L^{2}} = 0$ . Choose $g = (A - B) f$ , which is smooth and compactly supported on compact $S$ . The resulting integral $\int_{S} ∣ (A - B) f ∣^{2} d μ_{h}$ is zero, so strict positivity of the density integral forces $(A - B) f = 0$ everywhere. Hence $A = B$ , and the operator of step 3.1 is the formal adjoint $L^{†}$ .

5.1F1F6step 3.1step 4.1∎

The coordinate formulas give $Δ_{h} u = γ^{- 1} \partial_{i} (γ h^{i j} \partial_{j} u)$ , whose second-order part is $h^{i j} \partial_{i} \partial_{j} u$ . The drift and multiplication terms in $L$ and $L^{†}$ have order at most one, so both have second-order part $- h^{i j} \partial_{i} \partial_{j}$ . In the convention F6 their symbol is $h^{i j} ξ_{i} ξ_{j} = h^{- 1} (ξ, ξ) = ∣ ξ ∣_{h}^{2} > 0$ for nonzero $ξ$ . This is an intrinsic quadratic form, hence chart independent, and the calculation includes dimension one. This proves all assertions of the Statement without an orientation assumption.

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