Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Laurence H. Mayther

TL;DR

This work proves the relative $h$-principle for closed $\mathrm{SL}(3;\mathbb{R})^2$ 3-forms on oriented 6-manifolds by employing convex integration with avoidance. By reducing the problem to a fibred differential relation $\mathscr{R}_+(a)$, the authors navigate the non-ampleness obstacle through a novel generic-hyperplane analysis and the concept of macilence, establishing ampleness in a generic setting. Consequently, any manifold that admits an $\mathrm{SL}(3;\mathbb{R})^2$ 3-form represents every $3$-class with such a form, and the Hitchin functional on each class is unbounded above. These results extend the scope of stable forms known to satisfy the relative $h$-principle and illuminate the interaction between rank-3 distributions and hyperplane data in 6-manifolds, with implications for calibrated geometry and special holonomy.

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.

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

TL;DR

This work proves the relative -principle for closed 3-forms on oriented 6-manifolds by employing convex integration with avoidance. By reducing the problem to a fibred differential relation , the authors navigate the non-ampleness obstacle through a novel generic-hyperplane analysis and the concept of macilence, establishing ampleness in a generic setting. Consequently, any manifold that admits an 3-form represents every -class with such a form, and the Hitchin functional on each class is unbounded above. These results extend the scope of stable forms known to satisfy the relative -principle and illuminate the interaction between rank-3 distributions and hyperplane data in 6-manifolds, with implications for calibrated geometry and special holonomy.

Abstract

This paper uses convex integration with avoidance and transversality arguments to prove the relative -principle for closed 3-forms on oriented 6-manifolds. As corollaries, it is proven that if an oriented 6-manifold admits any 3-form, then every degree 3 cohomology class on can be represented by an 3-form and, moreover, that the corresponding Hitchin functional on 3-forms representing this class is necessarily unbounded above. Essential to the proof of the -principle is a careful analysis of the rank 3 distributions induced by an 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.
Paper Structure (4 sections, 3 theorems, 23 equations)

This paper contains 4 sections, 3 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $\hbox{\boldmath $\operatorname{SL}(3;\mathbb{R})^2$}$ 3-forms
  4. Lemma \ref{['mac-lem']}(2): the macilence of $\Sigma^\pm_{\{\mathbb{B}',\mathbb{B}"\}}$

Key Result

Theorem 1.1

$\operatorname{SL}(3;\mathbb{R})^2$ 3-forms satisfy the relative $h$-principle. In particular, taking $A = \emptyset$ in the definition of the relative $h$-principle, the inclusions: are homotopy equivalences and thus if $\mathrm{M}$ admits any $\operatorname{SL}(3;\mathbb{R})^2$ 3-form, then every degree 3 cohomology class on $\mathrm{M}$ can be represented by an $\operatorname{SL}(3;\mathbb{R})

Theorems & Definitions (5)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • proof
  • proof