Norms on the cohomology of hyperbolic 3-manifolds

• 成果类型: Article
• 署名作者: Brock, Jeffrey F.; Dunfield, Nathan M.
• 署名单位: Brown University; University of Illinois System; University of Illinois Urbana-Champaign
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-017-0735-3
• 发表日期: 2017
• 页码: 531-558
• 关键词: analytic torsion; SURFACES; HOMOLOGY; AREA

摘要

We study the relationship between two norms on the first cohomology of a hyperbolic 3-manifold: the purely topological Thurston norm and the more geometric harmonic norm. Refining recent results of Bergeron, Aengun, and Venkatesh as well as older work of Kronheimer and Mrowka, we show that these norms are roughly proportional with explicit constants depending only on the volume and injectivity radius of the hyperbolic 3-manifold itself. Moreover, we give families of examples showing that some (but not all) qualitative aspects of our estimates are sharp. Finally, we exhibit closed hyperbolic 3-manifolds where the Thurston norm grows exponentially in terms of the volume and yet there is a uniform lower bound on the injectivity radius.