Gluing local first integrals to a global function ψ under local admissibility constraints

Determine whether locally-defined functions ψ that integrate the Frobenius distribution span(u) ⊕ ker S and satisfy u·∇ψ = 0 together with π_parallel·∇ψ = 0 (where π_parallel is the orthogonal projector onto ker S) can be glued to produce a smooth globally-defined ψ on the toroidal annulus Q = S¹ × S¹ × [0,1] under only the local admissibility PDE conditions tr(S) = 0, det(S) = 0, and ℒ_u(π_parallel) = 0.

Background

From the local analysis, the rank-2 bundle span(u) ⊕ ker S is integrable, guaranteeing locally-defined functions ψ with u·∇ψ = 0 and π_parallel·∇ψ = 0. The global admissibility theorem introduces an explicit global condition (II.d) requiring the existence of a smooth ψ with nowhere-vanishing gradient satisfying these properties, reflecting a topological constraint.

The authors note that while local integrability holds by Frobenius, it is not evident that local ψ can be assembled into a single global function; they emphasize this as a topological question. An appendix example illustrates situations where a global ψ fails to exist, underscoring that additional conditions are needed to secure global integrability.

Clarifying when local data can be globalized would refine the sufficiency of the local admissibility constraints and delineate the necessary topological assumptions for constructing quasisymmetric fields on toroidal annuli.