Relative $h$-principles for closed stable forms

TL;DR

This paper addresses the problem of proving relative $h$-principles for closed, stable $p$-forms on manifolds and develops a convex-integration framework that yields relative $h$-principles for four new classes, namely $\widetilde{\mathrm{G}}_2$ 3-forms and 4-forms, ossymplectic forms, and ospseudoplectic forms. It further provides unified proofs of existing $h$-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, and shows that whenever a class satisfies the relative $h$-principle, the Hitchin functional is unbounded above. A key technical advance is an analogue of the Hodge decomposition on arbitrary manifolds (Appendix A) that enables convex-integration arguments in this general setting. Together, these results broaden the flexibility of stable forms to realize prescribed de Rham cohomology classes and unify several strands of calibrated geometry under a single convex-integration approach.

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.

Theorems & Definitions (5)

• Theorem 1.1: Theorems \ref{['os']}, \ref{['op']}, \ref{['sg-4']} and \ref{['sg-3']}
• Theorem 1.2
• Definition 1.3
• Lemma 1.4
• proof : Proof of Lemma \ref{['hP->hP']}