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Principal eigenfunctions and MOTS spectral stability

Statement

On a connected compact smooth Riemannian manifold $(S, h)$ without boundary, of positive dimension, let $L = - Δ_{h} + 2 t \cdot \nabla + c$ have smooth real coefficients. There is a real principal eigenvalue $λ$ , a smooth everywhere positive real eigenfunction $ϕ$ unique up to multiplication by a positive constant, and a smooth positive adjoint eigenfunction $χ$ at the same eigenvalue. Every other eigenvalue $μ$ satisfies $Re μ \geq λ$ .

For a connected closed smooth MOTS $S$ in the supplied smooth spacetime, fix smooth future-directed null normals $l, k$ with $g (l, k) = - 2$ and outgoing expansion $θ_{l} = 0$ . Here an admissible outward normal direction means a fixed smooth vector field $v$ normal to $S$ , pointwise linearly independent of $l$ , and normalized by $g (l, v) = 1$ . Its smooth elliptic stability operator $L_{v} ψ = δ_{ψ v} θ_{l}$ has $λ \geq 0$ if and only if the MOTS is stably outermost in that direction. Here this means that some smooth $ψ \geq 0$ , $ψ \equiv̸ 0$ , has $δ_{ψ v} θ_{l} \geq 0$ everywhere. Strict stability means additionally that this variation is positive somewhere; it is equivalent to $λ > 0$ . For the outward graph in a spacelike hypersurface use $v = ν$ , $l = n + ν$ and $k = n - ν$ . This spectral criterion by itself establishes no stability claim for a particular bifurcation sphere.

Facts & Assumptions

[F1]

Formal adjoint, smooth measure and real coefficients are fixed; the adjoint has the same elliptic leading symbol. Formal adjoint of a real drift Laplacian on a closed manifold The Axiom of Choice

[F2]

Positive right and adjoint eigenpairs exist by the explicitly imported closed-manifold theorem. Imported positive eigenpairs for a smooth closed elliptic operator

[F3]

Adjoint pairing identifies their eigenvalues, and the ground-state identity gives spectral ordering and uniqueness. Positive right and adjoint eigenfunctions have the same eigenvalue Ground-state identity, spectral ordering and eigenspace uniqueness

[F4]

Nonnegative supersolutions characterize the common eigenvalue sign. Nonnegative supersolutions detect the principal spectral sign

[F5]

The authored surface convention gives an elliptic outward graph operator and the exact null-pair normalization. Surface measure, flux densities and null normal gauge Outward graph stability has the negative-Laplacian principal symbol

[F6]

For the fixed smooth null gauge $g (l, k) = - 2$ and smooth normal field $v$ with $g (l, v) = 1$ , Andersson–Mars–Simon §3 equation (2), Definition 3.1 and Lemma 3.2 supply $δ_{ψ v} θ_{l} = L_{v} ψ$ and the smooth real negative-Laplacian form of $L_{v}$ . Their Definition 5.1 supplies the directional stability and strict-stability conventions used here. Andersson–Mars–Simon.

Proof

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

Goal: establish the Statement.

Proof technique: direct.

1.1F1F2F3

Apply the imported positive-eigenpair theorem to $L$ and its adjoint; their exact smooth real closed-manifold hypotheses are those in the Statement and in the formal-adjoint lemma. The adjoint-pairing lemma gives a common real eigenvalue. The ground-state identity shows that every smooth complex eigenvalue has real part at least this one and that its eigenspace consists of constant multiples of $ϕ$ . Rescaling $ϕ$ positively is exactly the ambiguity among everywhere positive real eigenfunctions. This proves the spectral assertions.

2.1F2F4F5F6step 1.1∎

For the MOTS application keep the direction and null gauge fixed as in the Statement. In the hypersurface graph convention the geometric helper proves $δ_{ψ ν} θ_{l} = L ψ$ , with $L = - Δ_{q} + 2 t \cdot \nabla + c$ having smooth real coefficients, so step 1.1 applies. Its null-pair normalization gives $g (l, k) = - 2$ and $g (l, ν) = 1$ . For every other admissible fixed direction, [F6] supplies $δ_{ψ v} θ_{l} = L_{v} ψ$ with $L_{v}$ of the same smooth real negative-Laplacian form, so step 1.1 applies to it as well. By the imported stability definition in [F6], stability is exactly existence of a nonzero smooth nonnegative $ψ$ with $L_{v} ψ \geq 0$ , and strict stability additionally requires positivity somewhere. Apply the spectral-sign lemma to obtain the equivalence with $λ \geq 0$ , and its strict version for $λ > 0$ . This is a criterion only; further geometry is required to verify either sign for any specific MOTS. This proves the Statement.

External input

The imported MOTS stability convention is Andersson–Mars–Simon Definition 5.1: a fixed direction $v$ , a nonzero smooth nonnegative test $ψ$ , and the pointwise sign of $δ_{ψ v} θ_{l}$ ; strictness requires positivity somewhere. Fix smooth future-directed null normals $l, k$ with $g (l, k) = - 2$ and a smooth normal field $v$ , pointwise independent of $l$ , with $g (l, v) = 1$ , as in their §3 equation (2). Their Definition 3.1 and Lemma 3.2 supply the smooth real negative-Laplacian operator $L_{v}$ and identify $δ_{ψ v} θ_{l} = L_{v} ψ$ for every such fixed direction. The hypersurface graph normalization and principal symbol are independently calculated in the geometric helper, and the sign equivalence is locally proved rather than assumed.

Source notes

This mathematical prerequisite uses the independent Andersson–Mars–Simon source for the normal-direction convention, general-direction variation operator and stability definition. The imported eigenpair theorem remains an explicitly external input; the local helpers do not purport to reconstruct the compact-resolvent existence proof.

Depends on

Used by · 3

Dependency tree · 2 levels

Each result is drawn at its shortest distance from this item. Arrows run from prerequisites to the items that use them; this item has a heavier outline. External source nodes link directly to the cited text. The view shows up to two dependency steps; open a prerequisite to continue.

External dependencies

Lars Andersson, Marc Mars and Walter Simon, Stability of marginally outer trapped surfaces and existence of marginally outer trapped tubes · §2 null-frame conventions, printed p.5; §3 variation definition, printed p.7, equation (2), Definition 3.1 equations (5) and Lemma 3.2 equation (7), printed p.8; §5 Definition 5.1 and Proposition 5.1 with proof, printed pp.14–15

For a smooth MOTS and a fixed smooth normal direction v with g(l,v)=1, stability means existence of smooth psi>=0, psi not identically zero, with delta_(psi v) theta_l>=0. Strict stability requires this variation to be positive somewhere. The variation equals L_v psi and Proposition 5.1 identifies these signs with the principal spectral sign.

Hypotheses: Connected closed smooth spacelike MOTS; fixed future null l,k with g(l,k)=-2; normal direction v independent of l and normalized g(l,v)=1; fixed normal gauge for variation. In a spacelike hypersurface v=nu is the chosen outward unit normal. No energy condition is required for this equivalence.

Reading limits: Consulted the arXiv PDF text for §2 null-frame conventions (printed p.5), §3 variation definition and normal-direction hypotheses (pp.7–8), Definition 3.1 and Lemma 3.2 (p.8), and §5 Definition 5.1 and Proposition 5.1 with proof (pp.14–15). Imports are the normalized direction class, general-direction elliptic variation formula and stability definition. The hypersurface graph principal symbol and spectral-sign implication have local helper proofs. The general variation formula is accepted as cited external input; its Appendix A derivation was not reconstructed. No stability of a particular surface is imported.

Sources

Lars Andersson, Marc Mars and Walter Simon, Stability of marginally outer trapped surfaces and existence of marginally outer trapped tubes · §2 null-frame conventions, printed p.5; §3 variation definition, printed p.7, equation (2), Definition 3.1 equations (5) and Lemma 3.2 equation (7), printed p.8; §5 Definition 5.1 and Proposition 5.1 with proof, printed pp.14–15