Essential minimal volume of Einstein 4-manifolds
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866916126992105472 |
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| author | Song, Antoine |
| author_facet | Song, Antoine |
| contents | The minimal volume of a closed manifold $M$ is the infimum of the volume of $(M,g)$ over all metrics $g$ with sectional curvature between $-1$ and $1$. We introduce a variant called the essential minimal volume, $\mathrm{ess-Minvol}(M)$, which is the limit, as $δ>0$ goes to $0$, of the infimum of the volume of the $δ$-thick part of $(M,g)$ over all metrics $g$ with sectional curvature between $-1$ and $1$. We show that, for some universal constant $C>0$, any closed Einstein 4-manifold $M$ with Euler characteristic $e(M)$ satisfies $$C^{-1}e(M) \leq \mathrm{ess-Minvol}(M) \leq Ce(M).$$ As a corollary, these inequalities are true for the essential minimal volume of closed complex surfaces of nonnegative Kodaira dimension. We conjecture that those linear bounds in fact hold for the minimal volume. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2103_05659 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Essential minimal volume of Einstein 4-manifolds Song, Antoine Differential Geometry Geometric Topology The minimal volume of a closed manifold $M$ is the infimum of the volume of $(M,g)$ over all metrics $g$ with sectional curvature between $-1$ and $1$. We introduce a variant called the essential minimal volume, $\mathrm{ess-Minvol}(M)$, which is the limit, as $δ>0$ goes to $0$, of the infimum of the volume of the $δ$-thick part of $(M,g)$ over all metrics $g$ with sectional curvature between $-1$ and $1$. We show that, for some universal constant $C>0$, any closed Einstein 4-manifold $M$ with Euler characteristic $e(M)$ satisfies $$C^{-1}e(M) \leq \mathrm{ess-Minvol}(M) \leq Ce(M).$$ As a corollary, these inequalities are true for the essential minimal volume of closed complex surfaces of nonnegative Kodaira dimension. We conjecture that those linear bounds in fact hold for the minimal volume. |
| title | Essential minimal volume of Einstein 4-manifolds |
| topic | Differential Geometry Geometric Topology |
| url | https://arxiv.org/abs/2103.05659 |