The relative $h$-principle for closed $\mathrm{SL}(3;\mathbb{R})^2$ 3-forms
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866912823317102592 |
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| author | Mayther, Laurence H. |
| author_facet | Mayther, Laurence H. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2309_15832 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The relative $h$-principle for closed $\mathrm{SL}(3;\mathbb{R})^2$ 3-forms Mayther, Laurence H. Geometric Topology Algebraic Topology Differential Geometry 53C10, 53C15, 15A69, 15A72, 57N75, 35R45, 58A30 (Primary) 58A20 (Secondary) 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. |
| title | The relative $h$-principle for closed $\mathrm{SL}(3;\mathbb{R})^2$ 3-forms |
| topic | Geometric Topology Algebraic Topology Differential Geometry 53C10, 53C15, 15A69, 15A72, 57N75, 35R45, 58A30 (Primary) 58A20 (Secondary) |
| url | https://arxiv.org/abs/2309.15832 |