LemmaStatement: AI-generatedProof: AI-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Nonnegative supersolutions detect the principal spectral sign

Statement

Let $(S, h)$ be a nonempty connected compact smooth Riemannian manifold without boundary, of positive dimension. Let $t$ be a smooth real vector field and $c$ a smooth real function on $S$ . Put $L u = - Δ_{h} u + 2 ⟨ t, \nabla u ⟩_{h} + c u$ , where $Δ_{h} = {div}_{h} {grad}_{h}$ . Use the positive Riemannian density $d μ_{h}$ and the formal adjoint $L^{†} u = - Δ_{h} u - 2 ⟨ t, \nabla u ⟩_{h} + (c - 2 {div}_{h} t) u$ for the pairing $⟨ f, g ⟩ = \int_{S} \overline{f} g d μ_{h}$ .

Assume smooth real functions $ϕ > 0$ and $χ > 0$ satisfy $L ϕ = λ ϕ$ and $L^{†} χ = λ χ$ for a real constant $λ$ . Then $λ \geq 0$ if and only if there is a smooth real $ψ \geq 0$ , $ψ \equiv̸ 0$ , with $L ψ \geq 0$ pointwise. The strict variant $λ > 0$ is equivalent to existence of such a $ψ$ with $L ψ$ additionally positive somewhere. Thus a nonnegative test function suffices; it is not necessary to assume the test is everywhere positive.

Facts & Assumptions

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

Goal: establish the Statement.

[F1]

For the operator, manifold and density in the Statement, the formal-adjoint identity is $\int_{S} f L g d μ_{h} = \int_{S} (L^{†} f) g d μ_{h}$ for smooth real $f, g$ . The operators have smooth real coefficients and map smooth real functions to smooth real functions. Formal adjoint of a real drift Laplacian on a closed manifold

[F2]

Positive right and adjoint eigenfunctions at real eigenvalues have the same eigenvalue; the Statement denotes this common value by $λ$ . Positive right and adjoint eigenfunctions have the same eigenvalue

[F3]

Compactly supported smooth density integration is linear, nonnegative on nonnegative densities and strictly positive on any nonzero nonnegative density, without an orientation assumption. Every smooth density on compact $S$ has compact support, so this applies to the density products below. The stated ZFC assumption includes AC; applying AC to countable families gives the countable-choice hypothesis ${A C}_{ω}$ required for this integration theorem. Orientation-free density integration and its properties The Axiom of Choice

[F4]

Smooth real-valued maps on smooth manifolds are continuous; apply this to $ψ$ and the smooth function $L ψ$ . Smooth maps are continuous

Proof

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

Goal: establish the Statement.

[F1] For the operator, manifold and density in the Statement, the formal-adjoint identity is $\int_{S} f L g d μ_{h} = \int_{S} (L^{†} f) g d μ_{h}$ for smooth real $f, g$ . The operators have smooth real coefficients and map smooth real functions to smooth real functions. Formal adjoint of a real drift Laplacian on a closed manifold

[F2] Positive right and adjoint eigenfunctions at real eigenvalues have the same eigenvalue; the Statement denotes this common value by $λ$ . Positive right and adjoint eigenfunctions have the same eigenvalue

[F3] Compactly supported smooth density integration is linear, nonnegative on nonnegative densities and strictly positive on any nonzero nonnegative density, without an orientation assumption. Every smooth density on compact $S$ has compact support, so this applies to the density products below. The stated ZFC assumption includes AC; applying AC to countable families gives the countable-choice hypothesis ${A C}_{ω}$ required for this integration theorem. Orientation-free density integration and its properties The Axiom of Choice

[F4] Smooth real-valued maps on smooth manifolds are continuous; apply this to $ψ$ and the smooth function $L ψ$ . Smooth maps are continuous

Proof technique: direct.

1.1given

If $λ \geq 0$ , take $ψ = ϕ$ . Then $ψ$ is smooth, everywhere positive and nonzero because $S$ is nonempty, and $L ψ = λ ϕ \geq 0$ . If $λ > 0$ , this is positive everywhere. This proves both forward implications.

2.1givenF1F2F3F4step 1.1∎

Conversely, let $ψ$ be a smooth real nonzero nonnegative function with $L ψ \geq 0$ . Adjoint integration gives $\int_{S} χ L ψ d μ_{h} = \int_{S} (L^{†} χ) ψ d μ_{h} = λ \int_{S} χ ψ d μ_{h}$ . The last integral is strictly positive: continuity and nonzero nonnegativity of $ψ$ give a nonempty open set where $ψ > 0$ , and $χ$ is everywhere positive, so $χ ψ d μ_{h}$ is a nonzero nonnegative smooth density. The left side is nonnegative, hence $λ \geq 0$ . If $L ψ$ is positive at one point, its smoothness makes it positive on a nonempty open set. Then $χ L ψ d μ_{h}$ is also a nonzero nonnegative smooth density, so the left side is strictly positive and $λ > 0$ . Together with step 1.1 this proves both equivalences.

Depends on

Used by · 1

Principal eigenfunctions and MOTS spectral stability

Dependency tree · 2 levels

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