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Macroscopic scalar curvature and areas of cycles | Geometric and Functional Analysis

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https://link.springer.com/article/10.1007/s00039-017-0417-8

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Macroscopic scalar curvature and areas of cycles - Geometric and Functional Analysis

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In this paper we prove the following. Let $${\Sigma}$$ Σ be an n–dimensional closed hyperbolic manifold and let g be a Riemannian metric on $${\Sigma \times \mathbb{S}^1}$$ Σ × S 1 . Given an upper bound on the volumes of unit balls in the Riemannian universal cover $${(\widetilde{\Sigma\times \mathbb{S}^1},\widetilde{g})}$$ ( Σ × S 1 ~ , g ~ ) , we get a lower bound on the area of the $${\mathbb{Z}_2}$$ Z 2 –homology class $${[\Sigma \times \ast]}$$ [ Σ × * ] on $${\Sigma \times \mathbb{S}^1}$$ Σ × S 1 , proportional to the hyperbolic area of $${\Sigma}$$ Σ . The theorem is based on a theorem of Guth and is analogous to a theorem of Kronheimer and Mrowka involving scalar curvature.

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Macroscopic scalar curvature and areas of cycles

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Geometric and Functional Analysis - 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...

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