Macroscopic scalar curvature and areas of cycles
| Autor: | Alpert, Hannah, Funano, Kei |
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| Rok vydání: | 2017 |
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| Druh dokumentu: | Working Paper |
| Popis: | In this paper we prove the following. Let $\Sigma$ be an $n$--dimensional closed hyperbolic manifold and let $g$ be a Riemannian metric on $\Sigma \times \mathbb{S}^1$. Given an upper bound on the volumes of unit balls in the Riemannian universal cover $(\widetilde{\Sigma\times \mathbb{S}^1},\widetilde{g})$, we get a lower bound on the area of the $\mathbb{Z}_2$--homology class $[\Sigma \times \ast]$ on $\Sigma \times \mathbb{S}^1$, proportional to the hyperbolic area of $\Sigma$. The theorem is based on a theorem of Guth and is analogous to a theorem of Kronheimer and Mrowka involving scalar curvature. Comment: 14 pages, 0 figures; revised to match final version accepted by GAFA |
| Databáze: | arXiv |
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