Failure of famous functional inequalities on Finsler manifolds: the influence of $S$-curvature
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866917079721967616 |
|---|---|
| author | Kristály, Alexandru Li, Benling Zhao, Wei |
| author_facet | Kristály, Alexandru Li, Benling Zhao, Wei |
| contents | The validity of functional inequalities on Finsler metric measure manifolds is based on three non-Riemannian quantities, namely, the reversibility, flag curvature and $S$-curvature induced by the measure. Under mild assumptions on the reversibility and flag curvature, it turned out that famous functional inequalities -- as Hardy inequality, Heisenberg--Pauli--Weyl uncertainty principle and Caffarelli--Kohn--Nirenberg inequality -- usually hold on forward complete Finsler manifolds with non-positive $S$-curvature, cf. Huang, Kristály and Zhao [Trans. Amer. Math. Soc., 2020]. In this paper however we prove that -- under similar assumptions on the reversibility and flag curvature as before -- the aforementioned functional inequalities fail whenever the $S$-curvature is positive. Accordingly, our results clearly reveal the deep dependence of functional inequalities on the $S$-curvature. As a consequence of these results, we establish analytic aspects of Finsler manifolds, e.g., if the flag curvature is non-positive, the Ricci curvature is bounded from below and the $S$-curvature is positive, then the reversibility turns out to be infinite. Further topological properties and examples are presented on general Funk metric spaces, where the $S$-curvature plays again a decisive role. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_05497 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Failure of famous functional inequalities on Finsler manifolds: the influence of $S$-curvature Kristály, Alexandru Li, Benling Zhao, Wei Differential Geometry Analysis of PDEs The validity of functional inequalities on Finsler metric measure manifolds is based on three non-Riemannian quantities, namely, the reversibility, flag curvature and $S$-curvature induced by the measure. Under mild assumptions on the reversibility and flag curvature, it turned out that famous functional inequalities -- as Hardy inequality, Heisenberg--Pauli--Weyl uncertainty principle and Caffarelli--Kohn--Nirenberg inequality -- usually hold on forward complete Finsler manifolds with non-positive $S$-curvature, cf. Huang, Kristály and Zhao [Trans. Amer. Math. Soc., 2020]. In this paper however we prove that -- under similar assumptions on the reversibility and flag curvature as before -- the aforementioned functional inequalities fail whenever the $S$-curvature is positive. Accordingly, our results clearly reveal the deep dependence of functional inequalities on the $S$-curvature. As a consequence of these results, we establish analytic aspects of Finsler manifolds, e.g., if the flag curvature is non-positive, the Ricci curvature is bounded from below and the $S$-curvature is positive, then the reversibility turns out to be infinite. Further topological properties and examples are presented on general Funk metric spaces, where the $S$-curvature plays again a decisive role. |
| title | Failure of famous functional inequalities on Finsler manifolds: the influence of $S$-curvature |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2409.05497 |