The relative $h$-principle for closed $\mathrm{SL}(3;\mathbb{R})^2$ 3-forms

TL;DR

This paper proves a flexible classification result for closed $ ext{SL}(3; ext{R})^2$ 3-forms on 6-manifolds using convex integration, revealing topological and functional properties of these forms.

Contribution

It establishes the relative $h$-principle for closed $ ext{SL}(3; ext{R})^2$ 3-forms, introducing new techniques like the concept of macilence for ampleness verification.

Findings

Any cohomology class can be represented by an $ ext{SL}(3; ext{R})^2$ 3-form.

The Hitchin functional on these forms is unbounded above.

The $h$-principle applies to all manifolds admitting such forms.

Abstract

This paper uses convex integration with avoidance and transversality arguments to prove the relative $h$-principle for closed $\mathrm{SL}(3;\mathbb{R})^2$ 3-forms on oriented 6-manifolds. As corollaries, it is proven that if an oriented 6-manifold $\mathrm{M}$ admits any $\mathrm{SL}(3;\mathbb{R})^2$ 3-form, then every degree 3 cohomology class on $\mathrm{M}$ can be represented by an $\mathrm{SL}(3;\mathbb{R})^2$ 3-form and, moreover, that the corresponding Hitchin functional on $\mathrm{SL}(3;\mathbb{R})^2$ 3-forms representing this class is necessarily unbounded above. Essential to the proof of the $h$-principle is a careful analysis of the rank 3 distributions induced by an $\mathrm{SL}(3;\mathbb{R})^2$ 3-form and their interaction with generic pairs of hyperplanes. The proof also introduces a new property of sets in affine space, termed macilence, as a method of verifying ampleness.