Relative $h$-principles for closed stable forms
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866917203451838464 |
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| author | Mayther, Laurence H. |
| author_facet | Mayther, Laurence H. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2309_08721 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Relative $h$-principles for closed stable forms Mayther, Laurence H. Differential Geometry Algebraic Topology Functional Analysis Geometric Topology 53C10, 53D15, 15A69, 15A72, 55P10, 35R45 (Primary) 15A75, 58A20, 46A04, 46A11, 58A12, 57R05 (Secondary) 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. |
| title | Relative $h$-principles for closed stable forms |
| topic | Differential Geometry Algebraic Topology Functional Analysis Geometric Topology 53C10, 53D15, 15A69, 15A72, 55P10, 35R45 (Primary) 15A75, 58A20, 46A04, 46A11, 58A12, 57R05 (Secondary) |
| url | https://arxiv.org/abs/2309.08721 |