All-slope minimizing foliations do not imply rigidity without additional conditions. Identity holonomy at every rational slope restores rigidity through a new finite-amplitude resonant-cell classification.
Geometric Analysis
Solved problem 4 from arXiv:2602.12896 in the 1 base-dimension case via counterexample, and proved an affirmative rigidity theorem assuming identity holonomy only at rational slopes.
We distinguish ordinary and identity-holonomy minimizing foliations for \(u''=-V_u(x,u)\) on \(\T^2\). Under the ordinary deck-invariant definition, \(V(u)=-\frac12\sin^4(\pi u)\) admits minimizing foliations of every rotation number although \(V_u\not\equiv0\); its zero-rotation holonomy is nontrivial. If, for each rational rotation, there exists a minimizing foliation with identity primitive holonomy, we prove \(V_u\equiv0\) by reducing to, and classifying, phase-flat gauge-fixed rational cells of a periodic quadratic Lagrangian. The phase-space completion is obtained directly from order, compactness, circle translation numbers, and area preservation.
A rectangle in the covering plane. Coloured edges will be identified in pairs.
Each labelled leaf closes in its primitive class. This is the hypothesis used by the theorem.