Relative $h$-principles for closed stable forms

TL;DR

This paper develops a convex integration method to establish relative h-principles for various closed, stable forms on manifolds, including new classes, and explores implications for Hitchin functionals, with a novel Hodge decomposition result.

Contribution

Introduces a general convex integration approach to prove relative h-principles for new classes of closed stable forms and provides a novel Hodge decomposition on arbitrary manifolds.

Findings

Proved relative h-principles for four previously unknown classes of stable forms.

Unified proofs for existing h-principles of certain stable forms.

Showed that the Hitchin functional is unbounded above if the h-principle holds.

Abstract

This paper uses convex integration to develop a new, general method for proving relative $h$-principles for closed, stable, exterior forms on manifolds. This method is applied to prove the relative $h$-principle for 4 classes of closed stable forms which were previously not known to satisfy the $h$-principle, $\textit{viz.}$ stable $(2k-2)$-forms in $2k$ dimensions, stable $(2k-1)$-forms in $2k+1$ dimensions, $\widetilde{\mathrm{G}}_2$ 3-forms and $\widetilde{\mathrm{G}}_2$ 4-forms. The method is also used to produce new, unified proofs of all three previously established $h$-principles for closed, stable forms, $\textit{viz.}$ the $h$-principles for closed stable 2-forms in $2k+1$ dimensions, closed $\mathrm{G}_2$ 4-forms and closed $\mathrm{SL}(3;\mathbb{C})$ 3-forms. In addition, it is shown that if a class of closed stable forms satisfies the relative $h$-principle, then the corresponding Hitchin functional (whenever defined) is necessarily unbounded above. Due to the general nature of the $h$-principles considered in this paper, the application of convex integration requires an analogue of Hodge decomposition on arbitrary $n$-manifolds (possibly non-compact, or with boundary) which cannot, to the author's knowledge, be found elsewhere in the literature. Such a decomposition is proven in Appendix A.