שמור ב:
| מחבר ראשי: | |
|---|---|
| פורמט: | Preprint |
| יצא לאור: |
2023
|
| נושאים: | |
| גישה מקוונת: | https://arxiv.org/abs/2309.08721 |
| תגים: |
הוספת תג
אין תגיות, היה/י הראשונ/ה לתייג את הרשומה!
|
תוכן הענינים:
- 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.