A rigidity theorem for Einstein $4$-manifolds with semi-definite sectional curvature, and its consequences
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866908285638017024 |
|---|---|
| author | Di Cerbo, Luca F. |
| author_facet | Di Cerbo, Luca F. |
| contents | Any oriented $4$-dimensional Einstein metric with semi-definite sectional curvature satisfies the pointwise inequality \[ \frac{|s|}{\sqrt{6}}\geq|W^+|+|W^-|, \] where $s$, $W^+$ and $W^-$ are respectively the scalar curvature, the self-dual and anti-self-dual Weyl curvatures. We give a complete characterization of closed $4$-dimensional Einstein metrics with semi-definite sectional curvature saturating this pointwise inequality. We then present further consequences of this circle of ideas, in particular to the study of the geography of non-positively curved closed Einstein and Kaehler-Einstein $4$-manifolds. In the Kaehler-Einstein case, we obtain a sharp Gromov-Lueck type inequality. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_09570 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A rigidity theorem for Einstein $4$-manifolds with semi-definite sectional curvature, and its consequences Di Cerbo, Luca F. Differential Geometry Geometric Topology Any oriented $4$-dimensional Einstein metric with semi-definite sectional curvature satisfies the pointwise inequality \[ \frac{|s|}{\sqrt{6}}\geq|W^+|+|W^-|, \] where $s$, $W^+$ and $W^-$ are respectively the scalar curvature, the self-dual and anti-self-dual Weyl curvatures. We give a complete characterization of closed $4$-dimensional Einstein metrics with semi-definite sectional curvature saturating this pointwise inequality. We then present further consequences of this circle of ideas, in particular to the study of the geography of non-positively curved closed Einstein and Kaehler-Einstein $4$-manifolds. In the Kaehler-Einstein case, we obtain a sharp Gromov-Lueck type inequality. |
| title | A rigidity theorem for Einstein $4$-manifolds with semi-definite sectional curvature, and its consequences |
| topic | Differential Geometry Geometric Topology |
| url | https://arxiv.org/abs/2503.09570 |